arcsine distribution Levy arcsine distribution continuous The arcsine distribution models the proportion of time that Brownian motion is positive. \( a \in (-\infty, \infty) \), location \( w \in (0, \infty) \), scale \( a = 0 \), \( w = 1 \) \(x \in [a, a + w] \) \(f(x) = \frac{1}{\pi\sqrt{(x - a) (a + w -x)}}\) \(x \in \{a, a + w\} \) \(F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x - a}{w}\right)\) \(Q(u) = a + w \sin^2\left(\frac{\pi}{2} u\right), \; u \in (0, 1)\) \( M(t) = e^{a t} \sum_{n=0}^\infty \left(\prod_{j=0}^{n-1} \frac{2 j + 1}{2 j + 2}\right) \frac{w^n t^n}{n!}, \; t \in (-\infty, \infty) \) \(a + \frac{1}{2} w\) \(\frac{1}{8} w^2\) \(0\) \(-\frac{3}{2}\) \(a + \frac{1}{2} w\) \(a + \frac{2 - \sqrt{2}}{4} w\) \(a + \frac{2 + \sqrt{2}}{4} w\) Derived by Paul Levy in 1939 as the distribution of proportion of time that Brownian motion is positive. arnold1980some Bernoulli distribution discrete The Bernoulli distribution governs an indicator random variable. \(p \in [0, 1]\), the probability of the event \( p = \frac{1}{2} \) \(\{0, 1\}\) \(f(x) = p^x (1 - p)^{1 - x}, \; x \in \{0, 1\}\) \(\lfloor 2 p \rfloor\) \(F(x) = (1 - p)^{1 - x}, \; x \in \{0, 1\}\) \(Q(u) = F^{-1}(u), \; u \in [0, 1]\) where \(F\) is the distribution function \(G(t) = 1 - p + p t, \; t \in (-\infty, \infty)\) \(M(t) = 1 - p + p e^t, \; t \in (-\infty, \infty)\) \(\varphi(t) = 1 - p + p e^{i t}, \; t \in (-\infty, \infty)\) \(\mu(n) = p, \; n \in \{0, 1, \ldots\}\) \(p\) \(p (1-p)\) \(\frac{1 - 2 p}{\sqrt{p (1 - p)}}\) \(\frac{1- 6 p + 6 p^2}{p (1 - p)}\) \(-(1 - p) \ln(1 - p) - p \ln(p)\) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function power series exponential Named for Jacob Bernoulli marshall1985family beta distribution continuous The beta distribution is used to model random proportions and probabilities. \(\alpha \in (0, \infty)\), the left shape parameter \(\beta \in (0, \infty)\), the right shape parameter \( \alpha = 1 \), \( \beta = 1 \) \((0, 1)\) \(f(x) = \frac{1}{B(\alpha, \beta)} x^{\alpha-1}(1 - x)^{\beta-1}, \; x \in (0, 1)\), where \( B \) is the beta function \(\frac{\alpha - 1}{\alpha + \beta - 2}; \; \alpha \in (1, \infty), \beta \in (1, \infty)\) \(F(x) = \frac{B(x; a, b)}{B(a, b)}, \; x \in (0, 1)\), where \( x \mapsto B(x; a, b) \) is the incomplete beta function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\), where \(F\) is the distribution function. \(1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}\) \(\frac{\alpha}{\alpha + \beta}\) \(\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\) \(\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}\) \(\frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)}\) \(\ln(B(\alpha, \beta)) - (\alpha - 1) \psi(\alpha) - (\beta - 1) \psi(\beta) + (\alpha + \beta - 2) \psi(\alpha + \beta)\) where \(\psi\) is the digamma function \(Q\left(\frac{1}{2}\right) \) \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function exponential mcdonald1995generalization beta general distribution beta generalized distribution generalized beta distribution continuous The generalized beta distribution is used to model random proportions and probabilities. It extends the (standard) beta distribution, supported on \([0, 1]\) to an arbitrary range \([L, R]\). \(\alpha \in (0, \infty)\), the left shape parameter \(\beta \in (0, \infty)\), the right shape parameter \(L \in (-\infty, \infty)\), the left limit of the support range \(R \in (L, \infty)\), the right limit of the support range \((L, R)\) R \\ \frac{\beta}{R - L} & \text{for } x == L \text{ and } \alpha == 1 \\ \infty & \text{for } x == L \text{ and } \alpha > 1 \\ 0 & \text{for } x == L \text{ and } \alpha > 1 \\ \frac{\alpha}{R - L} & \text{for } x == R \text{ and } \beta == 1 \\ \infty & \text{for } x == R \text{ and } \beta < 1 \\ 0 & \text{for } x == R \text{ and } \beta > 1 \\ \exp(\log\Gamma(\alpha + \beta) - (\log\Gamma(\alpha) + \log\Gamma(\beta)) - \log(R - L) + (\alpha-1)*\log(\frac{x-L}{R-L})+(\beta-1)*\log(\frac{R-x}{R-L})) & otherwise \end{cases}\) ]]> \(I_{\frac{x-L}{R}}(\alpha,\beta)\!\) \( \frac{\alpha * (R - L)}{\alpha + \beta}\) \( \frac{(R-L)*(R-L)*\alpha*\beta}{(\alpha+\beta)^2*(\alpha+\beta+1)} \) exponential mcdonald1995generalization inverse beta distribution continuous The inverse beta distribution, as the name suggests, is the inverse probability dsitribution of a Beta-distributed variable. \(\alpha \in (0, \infty)\), the left shape parameter \(\beta \in (0, \infty)\), the right shape parameter \((0, 1)\) exponential mcdonald1995generalization binomial distribution discrete The binomial distribution models the number of successes in a fixed number of independent trials each with the same probability of success. \(n \in \{1, 2, \ldots\}\), the number of trials \(p \in [0, 1]\), the probability of success \( n = 1, \; p = \frac{1}{2} \) \(\{0, 1, \ldots, n\}\) \(f(x) = {n \choose x} p^x (1 - p)^{n - x}, \; x \in \{0, 1, \ldots, n\}\) \(\lfloor (n + 1) p \rfloor\) \(F(x) = B(1 - p; n - x, x + 1), \; x \in \{0, 1, \ldots, n\}\) where \(B\) is the incomplete beta function \(Q(r) = F^{-1}(r), \; r \in [0, 1]\) where \(F\) is the distribution function \(G(t) = (1 - p + p t)^n, \; t \in (-\infty, \infty)\) \(M(t) = (1 - p + p e^t)^n, \; t \in (-\infty, \infty)\) \(\varphi(t) = (1 - p + p e^{i t})^n, \; t \in (-\infty, \infty)\) \(n p\) \(n p (1 - p)\) \(\frac{1 - 2 p}{\sqrt{n p (1 - p)}}\) \(\frac{1 - 6 p (1 - p)}{n p (1 - p)}\) \(\frac{1}{2} \log_2[2 \pi e n p (1 - p)] + O\left(\frac{1}{n}\right)\) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{3}\right)\) where \(Q\) is the quantile function power series exponential The binomial distribution is attributed to Jacob Bernoulli altham1978two beta-binomial distribution discrete The beta-binomial distribution arises when the success parameter in the binomial distribution is randomized and given a beta distribution. \(n \in \{1, 2, \ldots\}\), the number of trials \(a \in (0, \infty)\), the left beta parameter \(b \in (0, \infty)\), the right beta parameter \( n = 1, \; a = 1, \; b = 1 \) \(\{0, 1, \ldots, n\}\) \( f(x) = \binom{n}{x} \frac{a^{[x]} b^{[n - x]}}{(a + b)^{[n]}} \), \( x \in \{0, 1, \ldots, n\} \) where \( m^{[j]} \) denotes the rising power of \( m \) of order \( j \) \(F(x) = \sum_0^x f(t), \quad x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function \(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function \(_{2}F_{1}(-n, a; a + b; 1 - e^t) \) \(\frac{n a}{a + b} \) \(n \frac{n a b (a + b + n)}{(a + b)^2 (a + b + 1)}\) \(\frac{(a + b + 2 n)(b - a)}{(a + b + 2)} \sqrt{\frac{1 + a + b}{n a b (n + a + b)}}\) \(\frac{(a + b)^2 ( + 1 + b)}{n a b (a + b + 2)(a + b + 3)(a + b + n)} \left[(a + b)(a + b - 1 + 6 n) + 3 a b (n - 2) + 6 n^2 - \frac{3 a b n (6-n)}{a + b} - \frac{18 a b n^2}{(a + b)^2}\right]\) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function altham1978two beta-negative binomial distribution discrete The beta-negative binomial distribution arises when the success parameter in the negative binomial distribution is randomized and given a beta distribution. \(k \in \{1, 2, \ldots\}\), the number of trials \(a \in (0, \infty)\), the left beta parameter \(b \in (0, \infty)\), the right beta parameter \( k = 1, \; a = 1, \; b = 1 \) \(\{k, k + 1, \ldots \}\) \( f(x) = \binom{n - 1}{x - 1} \frac{a^{[x]} b^{[n-x]}}{(a + b)^{[n]}} \), where \( r^{[j]} \) denotes the rising power of order \( j \) \(F(x) = \sum_0^x f(t), \quad x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function \(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function \(k \frac{a + b - 1}{a - 1}\) if \( a \gt 1 \) \( k \frac{a + b - 1}{(a - 1)(a - 2)}[b + k (a + b - 2)] - k^2 \left(\frac{a + b - 1}{a - 1}\right)^2 \) if \( a \gt 2 \) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function johnson2005univariate Cauchy distribution Cauchy-Lorentz distribution Lorentz distribution Breit-Wigner distribution continuous The general Cauchy distribution is the location-scale family associated with the standard Cauchy distribution \(a \in (-\infty, \infty)\). the location parameter \(b \in (0, \infty)\), the scale parameter \(\displaystyle x \in (-\infty, \infty)\!\) \( a = 0, \; b = 1 \) \(\frac{1}{\pi b \, \left[1+\left(\frac{x - a}{b}\right)^2\right]}\!\) \( a \) \(\frac{1}{\pi}\arctan\left(\frac{x - a}{b} \right) + \frac{1}{2} \) \(Q(p) = F^{-1}(p) = a + b \tan \left(\pi (p - \frac{1}{2}) \right), \; p \in (0, 1)\) Does not exist \(\varphi(t) = \exp(a i t - b |t|), \; t \in (-\infty, \infty)\) Does not exist Does not exist Does not exist Does not exist \(\ln (4 \pi b ) \) \( a \) \(a - b\) \(a + b\) location scale The distribution was first used by Simeon Poisson in 1824 and was re-introduced by Augustin Cauchy in 1853. It is also named for Hendrick Lorentz. haas1970inferences chi-square distribution chi-squared distribution continuous The chi-square distribution governs the sum of squares of independent standard normal variable. \(n \in (0, \infty)\), degrees of freedom \( n = 1 \) \((0, \infty)\) \(\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\; x^{\frac{n}{2}-1} e^{-\frac{x}{2}}\,\) \(n - 2, \; n \in [2, \infty)\) \(\frac{1}{\Gamma\left(\frac{n}{2}\right)}\;\gamma\left(\frac{n}{2},\,\frac{x}{2}\right)\) \(Q(p) = F^{-1}(p), \; p \in [0, 1)\) where \(F\) is the distribution function \(M(t) = \frac{1}{(1 - 2 t)^{n/2}}, \; t \in (-\infty, \frac{1}{2})\) \(\frac{1}{(1 - 2 i t^{n/2})}, \; t \in (-\infty, \infty)\) \(n\) \(2 n\) \(\sqrt{8 / n}\,\) \(12 / n\) \(\frac{n}{2} + \ln[2 \Gamma(n / 2)] + (1 - n / 2) \psi(n / 2)\) \(\approx n \left(1 - \frac{2}{9n}\right)^3\) \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function exponential The chi-square distribution was first used by Karl Pearson in 1900. lancaster2005chi non-central chi-square distribution non-central chi-squared distribution continuous The non-central chi-square distribution distribution is a generalization of the chi-squared distribution, which arises in the power analysis of statistical tests where the null distribution is asymptotically a chi-squared distribution; important examples of such tests are the likelihood ratio tests. \(k \in (0, \infty)\), degrees of freedom \(\lambda \in (0, \infty)\), non-centrality parameter \(x \in [0; +\infty)\,\) \(\frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})\) \(F(x) = 1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)\), where \(Q_M(a,b)\) is the Marcum Q-function \(\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}, \) for \( 2t \lt 1\) \(\frac{1}{(1 - 2 i t^{n/2}}) \; t \in (-\infty, \infty)\) \(k+\lambda\) \(2(k+2\lambda)\) \(\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}\) \(\frac{12(k+4\lambda)}{(k+2\lambda)^2}\) exponential sankaran1959non chi distribution continuous The chi distribution governs the square root of a variable with the chi-square distribution. \(n \in \{1, 2, \ldots\}\), the degrees of freedom \(x\in[0;\infty)\) \(\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}\) \(\sqrt{k-1}\,\)for\(k\ge1\) \(P(k/2,x^2/2)\,\) \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(\mu(k) = \frac{2^{k/2} \Gamma[(n+k)/2)]}{\Gamma(n/2)}, \; n \in \{1, 2, \ldots\}\) where \(\Gamma\) is the gamma function \(\mu=\sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}\) \(\sigma^2=k-\mu^2\,\) \(\gamma_1=\frac{\mu}{\sigma^3}\,(1-2\sigma^2)\) \(\frac{2}{\sigma^2}(1-\mu\sigma\gamma_1-\sigma^2)\) \(\ln[\Gamma(n/2)] + \frac{1}{2} [n - \ln(2) - (n-1) \psi_0(n/2)]\) where \(\psi_0\) is the polygamma function \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function krishnaiah1963note continuous uniform distribution rectangular distribution continuous The continuous uniform distribution governs a point chosen at random from an interval. The continuous uniform distribution, aka rectangular distribution, is a family of probability distributions where all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. It is the maximum entropy probability distribution for a random variate X under no constraint other than that it is contained in the distribution's support. \(a \in (-\infty, \infty)\), location, the left endpoint \( w \in (0, \infty) \), scale, the width of the interval \(b = a + w\), the right endpoint \( a = 0, \; w = 1 \) \([a, b]\) \(f(x) = \frac{1}{b - a}, \; x \in [a, b]\) all \(x \in [a, b]\) \(F(x) = \frac{x - a}{b - a}, \; x \in [a, b]\) \(Q(p) = a + p (b - a). \; p \in [0, 1]\) \(M(t) = \frac{e^{t b} - e^{t a}}{t (b - a)}, \; t \in (-\infty, \infty)\) \(\varphi(t) = \frac{e^{i t b} - e^{i t a}}{i t (b - a)}, \; t \in (-\infty, \infty)\) \(\mu(t) = \frac{b^{t+1} - a^{t+1}}{(t + 1)(b - a)}, \; t \in (0, \infty)\) \(\frac{1}{2}(a + b)\) \(\frac{1}{12} (b - a)^2\) \(0\) \(-\frac{6}{5}\) \(\ln(b - a)\) \(\frac{1}{2}(a + b)\) \(\frac{3}{4} a + \frac{1}{4}b\) \(\frac{1}{4} a + \frac{3}{4} b\) location scale kuipers2006uniform discrete uniform distribution discrete The discrete uniform distribution governs a point chosen at random from a discrete interval. \(a \in (-\infty, \infty)\), location, the left endpoint \( h \in (0, \infty) \), scale, the step size \( n \in \{1, 2, \ldots\} \), the number of points \(b = a + (n - 1) h\), the right endpoint \( a = 0, \; h = 1 \) \(\{a, a + h, \ldots, a + (n - 1) h\}\) \( f(x) = \frac{1}{n}, \; x \in \{a, a + h, \ldots, a + (n - 1) h\}\) \(x \in \{a, a + h, \ldots, a + (n - 1) h\}\) \(F(x) = \frac{1}{n}\left(\left \lfloor\ \frac{x - a}{h} + 1\right \rfloor \right), \; x \in [a, b] \) \(Q(p) = a + \left(\lceil n p \rceil - 1\right) h, \; p \in (0, 1] \) \(M(t) = \frac{1}{n} e^{t a} \frac{1 - e^{n t h}}{1 - e^{t h}}, \; t \in (-\infty, \infty)\) \(\frac{1}{2}(a + b) = a + \frac{1}{2}(n - 1)h\) \(\frac{1}{12}(n^2 - 1) h^2 = \frac{1}{12}(b - a)(b - a + h)\) \(0\) \(-\frac{6(n^2 + 1)}{5(n^2 - 1)}\,\) \(\ln(n)\,\) \(Q\left(\frac{1}{2}\right) \) where \( Q \) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \( Q \) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \( Q \) is the quantile function location scale freund1967modern exponential distribution negative exponential distribution continuous The exponential distribution models the time between random points in the Poisson model. \(r \in (0, \infty)\), rate \( b = 1 / r \), scale \( r = 1 \) \([0, \infty)\) \(f(x) = r e^{-r x}, \; x \in [0, \infty)\) \(0\) \(F(x) = 1 - e^{-r x}, \; x \in [0, \infty)\) \(Q(p) = \frac{- \ln(1 - p)}{r}, \; p \in [0, 1)\) \(\left(1 - \frac{t}{\lambda}\right)^{-1}\,\) \(\frac{r}{r - i t}, \; t \in (-\infty, \infty)\) \(\frac{1}{r}\) \(\frac{1}{r^2}\) \(2\) \(6\) \(1 - \ln(r)\) \(\frac{\ln(2)}{r}\) \(\frac{\ln(4) - \ln(3)}{r}\) \(\frac{\ln(3)}{r}\) exponential scale The exponential distribution was named by Karl Pearson in 1895. siegrist2007exponential exponential-logarithmic distribution continuous The exponential-logarithmic distribution models failure times of devices with decreasing failure rate. \(p \in (0, 1)\), the shape parameter \(b \in (0, \infty)\), the scale parameter \( p = \frac{1}{2}, \; b = 1 \) \([0,\infty)\) \(f(x) = \frac{1}{-\ln p} \frac{\beta(1 - p) e^{-x/b}}{1 - (1 - p) e^{-x/b}}, \; x \in [0, \infty)\) \(0\) \(F(x) = 1 - \frac{\ln(1 - (1 - p) e^{-x/b})}{\ln p}, \; x in [0, \infty)\) \(Q(u) = b \ln\left(\frac{1 - p}{1 - p^{1 - u}}\right), \; u \in (0, 1)\) \(\mu(n) = -n! b^n \frac{L_{n+1}(1 - p)}{\ln(p)}, \; n \in \{0, 1, \ldots\}\) where \(L_{n+1}\) is the polylog function of order \(n + 1\) \(-b \frac{L_2(1 - p)}{\ln(p)}\) where \(L_2\) is the polylog function of order \(2\). \(-b^2 \frac{2 L_3(1 - p)}{ln(p)} - \frac{L_2^2(1 - p)}{b^2 \ln^2(p)}\) where \(L_n\) is the polylog function of order \(n\) \(b \ln(1 + \sqrt{p})\) \(b \ln\left(\frac{1 - p}{1 - p^{3/4}}\right)\) \(b \ln\left(\frac{1 - p}{1 - p^{1/4}}\right)\) scale tahmasbi2008two exponential power distribution generalized error distribution continuous The exponential power distribution is a family of symmetric, unimodal distributions that generalizes the normal and Laplace families. \(\mu \in (-\infty, \infty)\), the location parameter \(\alpha \in (0, \infty)\), the scale parameter \(\beta \in (0, \infty)\), the shape parameter \(x \in (-\infty; +\infty)\!\) \(f(x) = \frac{\beta}{2 \alpha \Gamma(1/\beta)} \exp\left[-\left(\frac{|x - \mu|}{\alpha}\right)^\beta\right], \; x \in (-\infty, \infty)\) where \(\Gamma\) is the gamma function \(\mu\) \(F(x) = \frac{1}{2} + \frac{\sgn(x - \mu)}{2 \Gamma (1 / \beta)} \gamma\left[\frac{1}{\beta}, \left(\frac{|x - \mu|}{\alpha}\right)^\beta\right], \; x \in (-\infty, \infty)\), where \(\Gamma\) is the gamma function and \(\gamma\) is the lower incomplete gamma function \(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function \(\mu\) \(\frac{\alpha^2 \Gamma(3/\beta)}{\Gamma(1/\beta)}\) where \(\Gamma\) is the gamma function \(0\) \(\frac{\Gamma(5/\beta) \Gamma(1/\beta)}{\Gamma^2(3/\beta)} - 3\) where \(\Gamma\) is the gamma function \(\frac{1}{\beta} - \log\left[\frac{\beta}{2 \alpha \Gamma(1/\beta)}\right]\) where \(\Gamma\) is the gamma function \(\mu\) \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function location scale zhu2009properties F-distribution Snedecor's F-distribution Fisher-Snedecor distribution continuous The F-distribution governs the ratio of independent, scaled chi-square variables. \(n \in (0, \infty)\), numerator degrees of freedom \(d \in (0, \infty)\), denominator degrees of freedom \( n = 1, \; d = 1 \) \([0, \infty)\) \( f(x) = \frac{\Gamma[(n + d)/2]}{\Gamma(n/2) \Gamma(d/2)} \left(\frac{n}{d}\right)^{n/2} \frac{x^{(n-2)/2}}{[1 + (n/d) x]^{(n+d)/2}}, \; x \in [0, \infty) \) \(\frac{n - 2}{n} \frac{d}{d + 2} \) for \(n \gt 2\) \(F(x) = \frac{B[n x / (n x + d), n/2, d/2]}{B(n/2, d/2)}, \; x \in [0, \infty)\), where \( B \) is the beta function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function Does not exist \(\frac{d}{d - 2}\) for \(d \gt 2\) \(\frac{2 d (n + d - 2)}{n (d - 2)^2 (d - 4)}\) for \(d \gt 4\) \(\frac{(2 n + d - 2) \sqrt{8(d - 4)}}{(d - 6)\sqrt{n (n + d - 2)}}\) for \(d \gt 6 \) \(\frac{20 d - 8 d^2 + d^3 + 44 n - 32 n d + 5 n d^2 - 22 n^2 - 5 n^2 d - 16}{n (d - 6)(d - 8)(n + d - 2)/12}\) for \( d \gt 8 \) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function The \(F\)-distribution was first derived by George Snedecor in 1934. The letter F was chosen as a tribute to Ronald Fisher. johnson2005univariate gamma distribution continuous The gamma distribution governs the arrival times in the Poisson model, and has many applications in statistics. \(k \in (0, \infty)\), the shape parameter \(b \in (0, \infty)\), the scale parameter \( k = 1, \; b = 1 \) \((0,\,\infty) \) \(f(x) = \frac{1}{\Gamma(k) b^k} x^{k - 1}e^{-x / b}, \; x \in (0, \infty) \) where \( \Gamma \) is the gamma function \((k - 1) b \) for \( k \gt 1 \) \( F(x) = \frac{1}{\Gamma(k)} \gamma\left(k, \frac{x}{b}\right) \) where \( \gamma \) is the incomplete gamma function \(Q(p) = F^{-1}(p)\) where \(F\) is the distribution function \(M(t) = \frac{1}{(1 - b t)^k}, \; t \in (-\infty, 1 / b)\) \(\varphi(t) = \frac{1}{(1 - i b t)^k}, \; t \in (-\infty, \infty)\) \( k b\) \( k b^2 \) \( \frac{2}{\sqrt{k}} \) \( \frac{6}{k} \) \(\ln (4 \pi b)\) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function scale exponential siegrist2007exponential geometric distribution discrete The geometric distribution models the trial number of the first success in a sequence of Bernoulli trials. \(p \in (0, 1]\), the success parameter \( p = \frac{1}{2} \) \(\{1, 2, 3, \ldots\}\) \(f(k) = p (1 - p)^{k - 1}, \; k \in \{1, 2, \ldots\}\) \(1\) \( F(k) = 1 - (1 - p)^k \) \(Q(u) = \left\lceil \frac{\ln(1 - u)}{\ln(1 - p)} \right\rceil, \; u \in [0, 1)\) \(P(t) = \frac{p t}{1 - (1 - p)t}, \; t \in \left(-\frac{1}{1 - p}, \frac{1}{1 - p}\right)\) \(M(t) = \frac{p e^t}{1 - (1 - p) e^t}, \; t \in (-\infty, -\ln(1 - p))\) \(\varphi(t) = \frac{p e^{i t}}{1 - (1 - p) e^{i t}}, \; t \in (-\infty, \infty)\) \(\frac{1}{p}\) \(\frac{1 - p}{p^2}\) \(\frac{2 - p}{\sqrt{1 - p}}\!\) \(6 + \frac{p^2}{1 - p}\) \(\frac{-(1 - p) \log_2 (1 - p) - p \log_2 p}{p} \) \(\left\lceil \frac{-\ln(2)}{\ln(1 - p)} \right\rceil\) \(\left\lceil \frac{\ln(3) - \ln(4)}{\ln(1 - p)} \right\rceil\) \(\left\lceil \frac{-\ln(4)}{\ln(1 - p)} \right\rceil\) power series exponential The geometric distribution was used very early in the history of probability, but the name has been attributed to William Feller in 1950. philippou1983generalized hypergeometric distribution discrete The hypergeometric distribution governs the number of objects of a given type when sampling without replacement from a multi-type population. \(N\), the population size \(m\), the number of type 1 objects in the population \(n\), the sample size \( \left\{\max{(0,\, n+m-N)},\, \dots,\, \min{(m,\, n )}\right\}\) \( f(x) = \frac{\binom{m}{x} \binom{N - m}{n - x}}{\binom{N}{n}}, \; x \in \left\{\max{(0,\, n+m-N)},\, \dots,\, \min{(m,\, n )}\right\} \) \(\left \lfloor \frac{(n + 1)(m + 1)}{N + 2} \right \rfloor\) \(1-{{{n \choose {k+1}}{{N-n} \choose {m-k-1}}}\over {N \choose m}} \,_3F_2\!\!\left[\begin{array}{c}1,\ k+1-m,\ k+1-n \\ k+2,\ N+k+2-m-n\end{array};1\right]\) \(\frac{{N-m \choose n} \scriptstyle{\,_2F_1(-n, -m; N - m - n + 1; e^{t}) } } {{N \choose n}} \,\!\) \(n \frac{m}{N}\) \( n \frac{m}{N} \frac{N - m}{N} \frac{N - n}{N - 1} \) \(\frac{(N - 2 m)\sqrt{N - 1}(N - 2 n)}{\sqrt{n m(N - m)(N - n)}(N - 2)}\) \(\left[ \frac{N^2 (N-1)}{n(N - 2)(N - 3)(N - n)}\right] \left[ \frac{N(N+1) - 6 N(N - n)}{m (N - m)} + \frac{3 n (N - n)(N + 6)}{N^2} - 6 \right]\) The hypergeometric distribution is very old, and was used by Jacob Bernoulli, Abraham DeMoivre, and others. The named was coined by H.T. Gonin in 1936. harkness1965properties hyperbolic secant distribution continuous The hyperbolic secant distribution is a symmetric, unimodal distribution but with larger kurtosis than the normal distribution. \( \mu \in (-\infty, \infty) \), the location parameter \( \sigma \in (0, \infty) \), the scale parameter \( \mu = 0, \; \sigma = 1 \) \((-\infty, \infty)\) \( f(x) = \frac{1}{2 \sigma} \sech\left[\frac{\pi}{2}\left(\frac{x - \mu}{\sigma}\right)\right], \; x \in (-\infty, \infty) \) \(\mu\) \( F(x) = \frac{2}{\pi} \arctan\left\{\exp\left[\frac{\pi}{2} \left(\frac{x - \mu}{\sigma} \right) \right]\right\}, \; x \in (-\infty, \infty) \) \(Q(p) = \mu + \sigma \frac{2}{\pi} \ln[\tan(\frac{\pi}{2} p)], \; p \in (0, 1)\) \(M(t) = e^{\mu t} \sec(\sigma t), \; t \in (-\frac{\pi}{1}, \frac{\pi}{2 \sigma})\) \(\mu\) \(\sigma^2\) \(0\) \(2\) \(\mu\) \( \mu + \sigma \frac{2}{\pi} \ln(\sqrt{2} - 1)\) \( \mu + \sigma \frac{2}{\pi} \ln(\sqrt{2} + 1)\) location scale harkness1968generalized Irwin-Hall distribution continuous The Irwin-Hall distribution governs the sum of \(n\) independent variables, each uniformly distributed on \([0, 1]\). \(n \in \{1, 2, \ldots\}\), the number of terms \( n = 1 \) \([0, n]\) \(f(x) = \frac{1}{2 (n - 1)!} \sum_{k=0}^n (-1)^k \binom{n}{k}\sgn(x - k)(x - k)^{n-1}, \; x \in [0, n]\) \( n/2 \) for \( n \ge 2 \) \( F(x) = \frac{1}{2} + \frac{1}{2 n!} \sum_{k=0}^n (-1)^k \binom{n}{k} \sgn(x - k) (x - k)^n, \; x \in [0, n]\) \(M(t) = \left(\frac{e^t - 1}{t}\right)^n, \; t \in (-\infty, \infty)\) \(\frac{n}{2}\) \(\frac{n}{12}\) \(\frac{n}{2}\) The Irwin-Hall distribution is named for Joseph Irwin and Phillip Hall who independently analyzed the distribution in 1927. hall1927distribution inverted beta distribution beta prime distribution beta distribution of the second kind continuous The inverted beta distribution is conjugate for the odds in the Bernoulli distribution \(\alpha \in (0, \infty)\), the first shape parameter \(\beta \in (0, \infty)\), the second shape parameter \( \alpha = 1, \; \beta = 1 \) \(x > 0\!\) \(f(x) = \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\!\) \(\frac{\alpha - 1}{\beta + 1}\) if \(\alpha \in [1, \infty)\) \(F(x) = \int_0^x f(t) dt, \; x \in (0, \infty)\) where \(f\) is the probability density function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(\frac{\alpha}{\beta - 1}\) if \(\beta \in (1, \infty)\) \(\frac{\alpha (\alpha + \beta - 1)}{(\beta - 2)(\beta - 1)^2}\) if \(\beta \in (2, \infty)\) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function mcdonald1995generalization Laplace distribution double exponential distribution continuous The Laplace distribution is a symmetric, unimodal distribution with tails that are fatter than those of the normal distribution \(a \in (-\infty, \infty)\), location \(b \in (0, \infty)\), scale \( a = 0, \; b = 1 \) \( (-\infty, \infty) \) \(f(x) = \frac{1}{2 b} \exp \left(-\frac{\left|x - a\right|}{b} \right), \; x \in (-\infty, \infty) \) \( a \) \( F(x) = \begin{cases} \frac{1}{2} \exp\left(\frac{x - a}{b}\right), & x \in (-\infty, a] \\ 1 - \frac{1}{2} \exp\left(-\frac{x - a}{b}\right), & x \in [a, \infty) \end{cases} \) \(Q(p) = a + b \ln(2 \min\{p, 1 - p\}), \; p \in (0, 1)\) \(M(t) = \frac{e^{a t}}{1 - b^2 t}, \; t \in (-\frac{1}{b}, \frac{1}{b})\) \(\varphi(t) = \frac{e^{a i t}}{1 + b^2 t}, \; t \in (-\infty, \infty)\) \( a \) \(2 b^2\) \( 0 \) \( 3 \) \(\log(2 e b)\) \( a \) \(a - b \ln(2)\) \(a + b \ln(2)\) location The Laplace distribution is named for Pierre Simon Laplace. kotz2001laplace Levy distribution van der Waals profile stable distribution continuous The Levy distribution is a stable distribution that has applications in spectroscopy. \(a \in (-\infty, \infty)\), the location parameter \(b \in (0, \infty)\), the scale parameter \( a = 0, \; b = 1 \) \((a, \infty)\) \( f(x) = \sqrt{\frac{b}{2 \pi}} \frac{1}{(x - a)^{3/2}} \exp\left[-\frac{b}{2 (x - a)}\right], \; x \in (a, \infty)\) \( a + \frac{1}{3} b\) \( F(x) = 2 \left[1 - \Phi\left(\sqrt{\frac{b}{x - a}}\right)\right], \; x \in (a, \infty) \) where \( \Phi \) is the standard normal distribution function \( F^{-1}(p) = a + \frac{b}{\left[\Phi^{-1}(1 - p / 2)\right]^2}, \; p \in [0, 1) \) where \( \Phi^{-1} \) is the standard normal quantile function \(\varphi(t) = \exp\left(i a t - \sqrt{-2 i b t}\right), \; t \in (-\infty, \infty)\) \(\infty\) \(\infty\) undefined undefined \(\frac{1}{2}[1 + 3 \gamma + \ln(16 \pi b^2)]\) where \(\gamma\) is Euler's constant \( a + b \left[\Phi^{-1}\left(\frac{3}{4}\right)\right]^{-2}\) where \(\Phi^{-1}\) is the standard normal quantile function \( a + b \left[\Phi^{-1}\left(\frac{7}{8}\right)\right]^{-2}\) where \(\Phi^{-1}\) is the standard normal quantile function \( a + b \left[\Phi^{-1}\left(\frac{5}{8}\right)\right]^{-2}\) where \(\Phi^{-1}\) is the standard normal quantile function location scale stable The Levy distribution is named for Paul Pierre Levy. barndorff2001levy Landau distribution continuous The Landau distribution is used in physics to describe the fluctuations in the energy loss of a charged particle passing through a thin layer of matter. This distribution is a special case of the stable Levy distribution with parameters (1, 1). \(\mu \in (-\infty, \infty)\), the location parameter \(c \in (0, \infty)\), the scale parameter \(1, \infty)\) \(f(x) = \sqrt{\frac{1}{2 \pi}} \frac{e^{-1/2(x - 1)}}{(x - 1)^{3/2}}, \; x \in (1, \infty)\) \(1 + \frac{1}{3}\) \(F(x) = \int_1^x f(t) dt, \; x \in \) where \(f\) is the probability density function landau1944energy logarithmic distribution logarithmic series distribution log-series distribution discrete The logarithmic distribution is sometimes used to model relative species abundance. \(p \in (0, 1)\), the shape parameter \( p = \frac{1}{2} \) \(\{1, 2, 3, \ldots\}\) \(f(k) = \frac{-1}{\ln(1 - p)} \frac{p^k}{k}, \; k \in \{1, 2, \ldots\}\) \(1\) \(F(k) = 1 + \frac{B(p; k + 1, 0)}{\ln(1 - p)}, \; k \in \{1, 2, \ldots\} \) where \( B \) is the incomplete beta function \(G(t) = \frac{\ln(1 - p t)}{\ln(1 - p)}, \; t \in (-\frac{1}{p}, \frac{1}{p})\) \(M(t) = \frac{\ln(1 - p e^t)}{\ln(1 - p)}, \; t \in (-\infty, -\ln(p))\) \(\varphi(t) = \frac{\ln(1 - p e^{i t})}{\ln(1 - p)}, \; t \in (-\infty, \infty)\) \(\frac{-1}{\ln(1 - p)} \frac{p}{1 - p}\!\) \(-p \frac{p + \ln(1 - p)}{(1 - p)^2 \ln^2(1 - p)}\!\) power series The logarithmic distribution was first derived by Ronald Fisher in 1943. fisher1943relation logistic distribution continuous The logistic distribution occurs in logistic regression. \(a \in (-\infty, \infty)\), the location parameter \(b \in (0, \infty)\), the scale parameter \( a = 0, \; b = 1 \) \((-\infty, \infty)\) \(f(x) = \frac{e^{-(x - a)/b}}{b \left(1 + e^{-(x - a)/b}\right)^2}, \; x \in (-\infty, \infty)\) \( a \) \(F(x) = \frac{1}{1 + e^{-(x - a)/b}}, \; x \in (-\infty, \infty)\) \(Q(p) = a + b \ln\left(\frac{p}{1 - p}\right), \; p \in (0, 1)\) \(M(t) = e^{a t} B(1 - b t, 1 + b t)\) where \(B\) is the beta function \( a \) \( \frac{\pi^2}{3} b^2 \) \( 0 \) \( 6/5 \) \(\ln(b) + 2\) \( a \) \(a - \ln(3) b\) \(a + \ln(3) b\) location scale Logistic regression was first used by D.R. Cox in 1958. balakrishnan1992handbook generalized logistic distribution skew logistic distribution continuous The generalized logistic distribution represents several different families of probability distributions. One family is called the skew-logistic distribution. Other families of distributions that have also been called generalized ogistic distributions include the shifted log-logistic distribution, which is a generalization of the log-logistic distribution. \(\alpha >0\), the location parameter \(\beta >0\), the scale parameter \((-\infty, \infty)\) \(f(x;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}\frac{\exp(-\beta x)} {(1+\exp(-x))^{\alpha+\beta}}\) balakrishnan2009continuous log-normal distribution log normal distribution lognormal distribution Galton distribution continuous The log-normal distribution models certain skewed variables. \(\mu \in (-\infty, \infty)\), the normal mean \(\sigma \in (0, \infty)\), the normal standard deviation \( e^\mu \), the scale parameter \( \mu = 0, \; \sigma = 1 \) \((0, \infty)\) \(f(x) = \frac{1}{\sigma x \sqrt{2 \pi}}\, \exp\left(-\frac{\left[\ln(x) - \mu\right]^2}{2 \sigma^2}\right), \; x \in (0, \infty) \) \(e^{\mu - \sigma^2}\) \( F(x) = \Phi\left[\frac{\ln(x) - \mu}{\sigma}\right], \; x \in (0, \infty)\) where \( \Phi \) is the standard normal distribution function \(F^{-1}(p) = \exp\left[\mu + \sigma \Phi^{-1}(p)\right]\), where \(\Phi\) is the standard normal distribution function \(\mu(n) = \exp(\mu n + \frac{1}{2} \sigma^2 n^2), \; n \in \{0, 1, \ldots\}\) \(e^{\mu + \sigma^2/2}\) \((e^{\sigma^2} - 1) e^{2 \mu + \sigma^2}\) \((e^{\sigma^2} + 2) \sqrt{e^{\sigma^2} - 1}\) \(e^{4 \sigma^2} + 2 e^{3 \sigma^2} + 3e^{2 \sigma^2} - 6\) \(\frac12 + \frac12 \ln(2 \pi \sigma^2) + \mu\) \(e^{\mu}\,\) \(\exp\left[\mu + \sigma \Phi^{-1}\left(\frac{1}{4}\right)\right]\), where \(\Phi\) is the standard normal distribution function \(\exp\left[\mu + \sigma \Phi^{-1}\left(\frac{3}{4}\right)\right]\), where \(\Phi\) is the standard normal distribution function scale exponential The lognormal distribution was first studied by Donald McAlister in 1879, in response to a problem posed by Francis Galton. This historical origin is the reason for the alternative name Galton distribution. The term lognormal distribution was first used by J.H. Gaddum in 1945. famoye1995continuous log-logistic distribution Fisk distribution continuous The log-logistic distribution models lifetimes of devices whose failure rates at first increase and then decrease. \(k \in (0, \infty)\), the shape parameter \(b \in (0, \infty)\), the scale parameter \([0, \infty)\) \( k = 1, \; b = 1 \) \( f(x) = \frac{b^k k x^{k - 1}}{(b^k + x^k)^2}, \; x \in [0, \infty)\) \( b \left(\frac{k - 1}{k + 1}\right)^{1/k} \) if \( a \gt 1 \) \(F(x) = \frac{x^k}{b^k + x^k}, \; x \in [0, \infty)\) \(F^{-1}(p) = b \left(\frac{p}{1 - p}\right)^{1/k}, \; p \in [0, 1)\) \(\mu(n) = b^n \frac{\pi n / k}{\sin(\pi n / k)}, \; n \lt k\) \( \infty \) if \( 0 \lt k \le 1 \); \(b \frac{\pi / k}{\sin(\pi / k)}\) if \( k \gt 1 \) does not exist if \( 0 \lt k \le 1 \); \( \infty \) if \( 1 \lt k \le 2 \); \( b^2 \left[\frac{2 \pi / k}{\sin(2 \pi / k)} - \frac{\pi^2 / k^2}{\sin^2(\pi / k)}\right] \) if \( k \gt 2 \) \( b\) \( b (1/3)^{1/k} \) \(b 3^{1/k} \) scale The log-logistic distribution is known as the Fisk distribution by economists. P.R. Fisk used the distribution to model income in 1961. shoukri1988sampling Maxwell-Boltzmann distribution Maxwell distribution continuous The Maxwell-Boltzmann Distribution arises in the kinetic theory of gases. \(b \in (0, \infty)\), the scale parameter \( b = 1 \) \([0, \infty)\) \(f(x) = \frac{1}{b^3} \sqrt{\frac{2}{\pi}} x^2 \exp\left(-\frac{x^2}{2 b^2}\right), \; x \in [0, \infty)\) \(\sqrt{2} b\) \(F(x) = 2 \Phi\left(\frac{x}{b}\right) - \frac{1}{b} \sqrt{\frac{2}{\pi}} x \exp\left(-\frac{x^2}{2 b^2}\right) - 1\) where \( \Phi \) is the standard normal distribution function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(2 b \sqrt{\frac{2}{\pi}}\) \(\frac{b^2 (3 \pi - 8)}{\pi}\) \(\frac{2 \sqrt{2}(16 - 5 \pi)}{(3 \pi - 8)^{3/2}}\) \(4 \frac{(-96 + 40 \pi - 3 \pi^2)}{(3 \pi - 8)^2}\) \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function scale The Maxwell-Boltzman distribution is named for James Clerk Maxwell and Ludwig Boltzmann for their use of the distribution is modeling the energy of molecules in a gas. laurendeau2005statistical negative binomial distribution Pascal distribution discrete The negative binomial distribution governs the number of trials needed for a specified number of successes in the Bernoulli trials model. \(k \in \{1, 2, \ldots\}\), the number of successes \(p \in (0, 1]\), the success parameter \( k = 1, \; p = \frac{1}{2} \) \(\{k, k+1, \ldots\}\) \(f(x) = \binom{x - 1}{k - 1} p^x (1 - p)^{x-k}, \; x \in \{k, k+1, \ldots\}\) \(\lfloor 1 + \frac{k-1}{p}\rfloor\) \(F(x) = \sum_{j=k}^x f(j) , \; x \in \{k, k+1, \ldots\}\) where \(f\) is the probability density function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(G(t) = \left[\frac{p t}{1 - (1-p) t}\right]^k, \; t \in (-\frac{1}{1-p}, \frac{1}{1-p})\) \(M(t) = \left[\frac{p e^t}{1 - (1-p) e^t}\right]^k, \; t \in (-\infty, -\ln(1 - p))\) \(\varphi(t) = \left[\frac{p e^{i t}}{1 - (1-p) e^{i t}}\right]^k, \; t\in (-\infty, \infty)\) \(k \frac{1}{p}\) \(k \frac{1-p}{p^2}\) \(\frac{2-p}{\sqrt{k (1-p)}}\) \(\frac{1}{k} \left[6 + \frac{p^2}{1 - p}\right]\) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function The alternative name Pascal distribution is in honor of Blaise Pascal who used the distribution in his solution to the Problem of Points. el2006negative normal distribution Gaussian distribution error distribution continuous The normal distribution is used to model physical quantities that are subject to numerous small, random errors. \(\mu \in (-\infty, \infty)\), the location parameter \(\sigma \in (0, \infty)\), the scale parameter \( \mu = 0, \; \sigma = 1 \) \((-\infty, \infty)\) \(f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left[-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2 \right], \; x \in (-\infty, \infty)\) \(\mu\) \(F(x) = \Phi\left(\frac{x - \mu}{\sigma}\right), \; x \in (-\infty, \infty)\) where \(\Phi\) is the standard normal distribution function \(Q(p) = \mu + \sigma \Phi^{-1}(p), \; p \in (0, 1)\) where \(\Phi\) is the standard normal distribution function \(M(t) = \exp\left(\mu t + \frac{1}{2} \sigma^2 t^2\right), \; t \in (-\infty, \infty)\) \(\varphi(t) = \exp\left(i \mu t - \frac{1}{2} \sigma^2 t^2\right), \; t \in (-\infty, \infty)\) \(\mu\) \(\sigma^2\) \(0\) \(0\) \(\frac{1}{2} \ln(2 \pi e \sigma^2)\) \(\mu\) \(\mu - \Phi^{-1}\left(\frac{1}{4}\right) \sigma\) where \(\Phi\) is the standard normal distribution function \(\mu + \Phi^{-1}\left(\frac{1}{4}\right) \sigma\) where \(\Phi\) is the standard normal distribution function location scale exponential stable dinov2008central The normal distribution was first derived by Carl Friedrich Gauss in 1809 (hence the alternative name Gaussian distribution). The normalizing constant and the first version of the Central Limit Theorem were contributions by Pierre Simon Laplace. The term normalizing constant was popularized by Karl Pearson around the turn of the 20th century. students t distribution Student's t distribution t distribution continuous The t distribution arises when estimating the mean of a normally distributed population when the sample size is small and population standard deviation is unknown. It comes into play in various statistical analyses like Students t-test for assessing the between-group statistical significant differences of two sample means, construction of confidence intervals for difference between two population means, linear regression analyses, etc. Like the normal distribution, the t-distribution is symmetric, bell-shaped and unimodal. However it has heavier tails, meaning that it is more prone to producing values that fall far from its mean. The Students t-distribution is a special case of the generalized hyperbolic distribution \(n \in (0, \infty)\), degrees of freedom \( n = 1 \) \((-\infty, \infty)\) \(f(x) = \frac{\Gamma \left(\frac{n + 1}{2} \right)} {\sqrt{n \pi} \Gamma \left(\frac{n}{2} \right)} \left(1 + \frac{x^2}{n} \right)^{-\frac{n + 1}{2}}, \; x \in (-\infty, \infty)\) where \( \Gamma \) is the gamma function \(0\) \(F(x) = \frac{1}{2} + x \Gamma \left( \frac{n + 1}{2} \right) \frac{\,_2F_1 \left ( \frac{1}{2},\frac{n + 1}{2};\frac{3}{2}; - \frac{x^2}{n} \right)} {\sqrt{\pi n}\,\Gamma \left(\frac{n}{2}\right)}, \; x \in (-\infty, \infty)\), where \({ }_2F_1\) is the hypergeometric function 0 for \( n \gt 1 \) \(\frac{n}{n - 2}, \) for \( n \gt 2 \) 0 for \( n \gt 3 \) \(\frac{6}{n - 4}, \) for \( n \gt 4\) \(\frac{n + 1}{2}\left[\psi \left(\frac{1 + n}{2} \right) - \psi \left(\frac{n}{2} \right) \right] + \log{\left[\sqrt{n} B \left(\frac{n}{2}, \frac{1}{2} \right)\right]} \) 0 exponential li1957student truncated normal distribution continuous The truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below, above or on both sides. The truncated normal distribution has wide applications in statistics and econometrics \(\mu \in (-\infty, \infty)\), the location parameter \(\sigma \in (0, \infty)\), the scale parameter \(a \in (-\infty, \infty)\), left limit \(b \in (a, \infty)\), right limit \([a, b]\) \(f(x;\mu,\sigma,a,b)=\frac{1}{\sigma Z}\phi(\xi)\) b\end{array}\right.\) ]]> \(F(x;\mu,\sigma,a,b)=\frac{\Phi(\xi)-\Phi(\alpha)}{Z}\) \(\mu+\frac{\phi(\alpha)-\phi(\beta)}{Z}\sigma\) \(\sigma^2\left[1+\frac{\alpha\phi(\alpha)-\beta\phi(\beta)}{Z}-\left(\frac{\phi(\alpha)-\phi(\beta)}{Z}\right)^2\right]\) kotz2000continuous Pareto distribution Bradford distribution continuous The Pareto distribution models highly skewed variables that sometimes arise in economics. \(a \in (0, \infty)\), the shape parameter \(b \in (0, \infty)\), the scale parameter \( a = 1, \; b = 1 \) \([b, \infty)\) \(f(x) = \frac{a b^a}{x^{a+1}}, \; x \in [b, \infty)\) \(b\) \(F(x) = 1 - \left(\frac{b}{x}\right)^a, \; x \in [b, \infty)\) \(F^{-1}(p) = \frac{b}{(1 - p)^{1/a}}, \; p \in [0, 1)\) \(\varphi(t) = a (-i b t)^a \gamma(-a, -i b t)\) where \(\gamma\) is the lower incomplete gamma function \(\mu(n) = b^n \frac{a}{a - n}, \; n \in (0, k)\) \( b \frac{a}{a - 1} \) for \( a \gt 1 \) \( b^2 \frac{a}{(a - 1)^2 (a - 2)} \) for \( a \gt 2 \) \(\frac{2(1 + a)}{a - 3} \sqrt{\frac{a - 2}{a}}\) for \(a \gt 3\) \(\frac{6(a^3 + a^2 - 6 a - 2)}{a(a - 3)(a - 4)}\) for \(a \gt 4 s\) \(\ln\left(\frac{b}{a}\right) + \frac{1}{a} + 1\) \(b 2^{1/a}\) \(b \left(\frac{4}{3}\right)^{1/a}\) \(b 4 ^{1/a}\) scale The Pareto distributin is named for the Italian economist Vilfredo Pareto, who used the distribution to model wealth, income and other economic variables. arnold1985pareto Poisson distribution discrete The Poisson distribution models the number of random points in a region of time or space under certain ideal conditions. \(\lambda \in (0, \infty)\), the shape parameter \( \lambda = 1 \) \(\{0, 1, 2, \ldots\}\) \(f(k) = e^{-\lambda} \frac{\lambda^k}{k!}, \; k \in \{0, 1, \ldots\} \) \(\lfloor\lambda\rfloor\) if \( \lambda \) is not an integer; \( \lambda \) and \( \lambda - 1 \) if \( \lambda \) is a positive integer \(F(x) = \frac{1}{x!} \gamma(x + 1, \lambda), \; x \in \{0, 1, 2, \ldots\}\) where \(\gamma\) is the lower incomplete gamma function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(G(t) = e^{\lambda (t - 1)}, \; t \in (-\infty, \infty)\) \(M(t) = \exp\left(\lambda(e^t - 1)\right), \; t \in (-\infty, \infty)\) \(\varphi(t) = \exp\left(\lambda(e^{i t} - 1)\right), \; t \in (-\infty, \infty)\) \(m(k) = \lambda^k, \; k \in \{0, 1, 2, \ldots\}\) \(\lambda\) \(\lambda\) \(\sqrt{\lambda}\) \(\frac{1}{\lambda}\) \(\lambda [1 - \log(\lambda)] + e^{-\lambda} \sum_{k=0}^\infty \frac{\lambda^k \log(k!)}{k!}\) \(Q\left(\frac12\right) \approx \lfloor\lambda + 1/3 - 0.02/\lambda\rfloor\) \(Q\left(\frac14)\right)\) where \(Q\) is the quantile function \(Q\left(\frac34\right)\) where \(Q\) is the quantile function exponential power series The Poisson distribution is named for Simeon Poisson who first used the distribution in 1838 in a study of judgements in court cases. consul1973generalization Rademacher distribution discrete The Rademacher distribution arises in physics and in bootstrapping. \(k\in\{-1,1\}\,\) N/A \(Q(p) = -1, \; p \in [0, \frac{1}{2}]; \quad Q(p) = 1, \; p \in (\frac{1}{2}, 1]\) \(M(t) = \cosh(t), \; t \in (-\infty, \infty)\) \(M(t) = \cos(t), \; t \in (-\infty, \infty)\) \(\mu(n) = 1, \; n \in \{0, 2, \ldots\}; \quad \mu(n) = 0, \; n \in \{1, 3, \ldots\}\) \(0\,\) \(1\,\) \(0\,\) \(-2\,\) \(\ln(2)\) \(0\,\) \(-1\) \(1\) The Rademacher distribution is named for the German mathematician Hans Rademacher. montgomery1990distribution Rayleigh distribution continuous The Rayleigh distribution governs the magnitude of a vector with independent, normal components that have zero mean and the same variance. \(\sigma \in (0, \infty)\), scale \( \sigma = 1 \) \([0, \infty)\) \(f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2 \sigma^2}\right), \; x \in [0, \infty) \) \(\sigma\) \(F(x) = 1 - \exp\left(-\frac{x^2}{2 \sigma^2}\right), \; x \in [0, \infty)\) \(Q(p) = \sigma \sqrt{-2 \ln(1 - p)}, \; p \in [0, 1)\) \(M(t) = 1 + \sqrt{2 \pi} \sigma t \exp\left(\frac{\sigma^2 t^2}{2}\right) \Phi(t), \; t \in (-\infty, \infty)\) where \(\Phi\) is the standard normal distribution function \(\mu(n) = \sigma^n 2^{n/2} \Gamma\left(1 + \frac{n}{2}\right), \; n \in [0, \infty)\) \(\sigma \sqrt{\frac{\pi}{2}}\) \(\sigma^2 (2 - \pi / 2)\) \(\frac{2 \sqrt{\pi} (\pi - 3)}{(4 - \pi)^{3/2}}\) \(-\frac{6 \pi^2 - 24 \pi + 16}{(4 - \pi)^2}\) \(1 + \ln\left(\frac{\sigma}{\sqrt{2}}\right) + \frac{\gamma}{2}\) where \(\gamma\) is Euler's constant \(\sigma \sqrt{\ln(4)}\,\) \(\sigma \sqrt{\ln(16) - \ln(9)}\) \(\sigma \sqrt{\ln(16)}\) scale The Rayleigh distribution is named for the English mathematician Lord Rayleigh (John William Strutt). kuruoglu2004modeling Rice distribution Rician distribution continuous The Rice distribution governs the magnitude of a circular bivariate normal random vector. \(\nu \in [0, \infty)\), the distance parameter \(\sigma \in (0, \infty)\), the scale parameter \([0, \infty)\) \(\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)} {2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)\) \(F(x) = 1 - Q\left(\frac{\nu}{\sigma}, \frac{x}{\sigma}\right), \; x \in [0, \infty)\) where \(Q\) is the Marcum \(Q\)-function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(\sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)\) \(2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)\) \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function scale The Rice distribution is named for Stephen O. Rice who used the distribution in 1945 in his study of random noise. rice1945mathematical semicircle distribution Wigner distribution Stato-Tate distribution continuous The semicircle distribution arises as the limiting distribution of the eigenvalues of random symmetric matrices. \(r \in (0, \infty)\), scale (radius) \( a \in (-\infty, \infty) \), location (center) \( a = 0, \; r = 1 \) \( [a - r, a + r] \) \(f(x) = \frac{2}{\pi r^2} \sqrt{r^2 - (x - a)^2}, \; x \in [a - r, a + r]\) \( a \) \( F(x) = \frac{1}{2} + \frac{x - a}{\pi r^2} \sqrt{r^2 - (x - a)^2} + \frac{1}{\pi} \arcsin\left(\frac{x - a}{r}\right), \; x \in [a - r, a + r] \) \(Q(p) = F^{-1}(p), \; p \in [0, 1]\) where \(F\) is the distribution function \(M(t) = 2 e^{t a} \frac{I_1(r t)}{r t}, \; t \in (-\infty, \infty)\) where \(I_1\) is the modified Bessel function \(\varphi(t) = 2 e^{i t a} \frac{J_1(r t)}{r t}, \; t \in (-\infty, \infty)\) where \(J_1\) is the Bessel function For \( a = 0 \), \( \mu(2 n) = \left(\frac{r}{2}\right)^{2 n} \frac{1}{n + 1} \binom{2n}{n} \) and \( \mu(2 n + 1) = 0 \) for \( n \in \{0, 1, \ldots\} \) \( a \) \(\frac{r^2}{4}\) \(0\,\) \(-1\,\) \(\ln(\pi r) - \frac{1}{2}\) \( a\,\) \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function location scale The semicircle distribution was used by the physicist Eugene Wigner in the study of random matrices. The distribution was also used by Nikio Sato and John Tate in a conjecture in number theory. abramowitz1972handbook triangular distribution triangle distribution continuous The triangular distribution arises from various simple combinations of continuous uniform distributions. \(a \in (-\infty, \infty)\), location, the left endpoint \( w \in (0, \infty) \), scale, the width of the interval \( p \in [0, 1] \), shape \(b = a + w\), the right endpoint \(c = a + p w\), the location of the vertex \( a = 0, \; w = 1, \; p = \frac{1}{2} \) \([a, a + w]\) \(f(x) = \begin{cases} \frac{2(x - a)}{(b - a)(c-a)} & a \le x \leq c, \\ \frac{2(b - x)}{(b - a)(b - c)} & c \lt x \le b \end{cases} \) \( c \) \(F(x) = \begin{cases} \frac{(x - a)^2}{(b - a)(c - a)} & a \le x \leq c, \\ 1 - \frac{(b - x)^2}{(b - a)(b - c)} & c \lt x \le b \end{cases}\) \(Q(u) = \begin{cases} a + \sqrt{(b - a)(c - a) u}, & \; u \in [0, (c - a) / (b - a)] \\ b - \sqrt{(1 - u)(b - a)(b - c)}, & u \in [(c - a) / (b - a), 1] \end{cases}\) \(M(t) = 2 \frac{(b - c) e^{a t} - (b - a) e^{c t} + (c - a) e^{b t}}{(b - a)(c - a)(b - c) t^2}, \; t \in (-\infty, \infty)\) \(\frac{a + b + c}{3}\) \(\frac{a^2 + b^2 + c^2 - a b - a c - b c}{18}\) \(\frac{\sqrt2(a + b - 2 c)(2 a - b - c)(a - 2 b + c)}{5 (a^2 + b^2 + c^2 - a b - a c - b c)^{3/2}}\) \(-\frac{3}{5}\) \(\frac{1}{2} + \ln\left(\frac{b - a}{2}\right)\) \(\begin{cases} a + \frac{\sqrt{(b - a)(c - a)}}{\sqrt{2}}, & c \ge \frac{a + b}{2}, \\ b - \frac{\sqrt{(b - a)(b - c)}}{\sqrt{2}}, & c \le \frac{a + b}{2} \end{cases}\) \(\begin{cases} a + \sqrt{\frac{1}{4}(b - a)(c - a)}, & c \geq \frac{3}{4} a + \frac{1}{4} b \\ b - \sqrt{\frac{3}{4}(b - a)(b - c)} & c \leq \frac{3}{4} a + \frac{1}{4} b \end{cases}\) \(\begin{cases} a + \sqrt{\frac{3}{4}(b - a)(c - a)}, & c \geq \frac{1}{4} a + \frac{3}{4} b \\ b - \sqrt{\frac{1}{4}(b - a)(b - c)}, & c \leq \frac{1}{4} a + \frac{3}{4} b \end{cases}\) location scale ren2002novel U-quadratic distribution continuous the U-quadratic distribution models certain symmetric, bimodal variables. \(c \in (-\infty, \infty)\), location, center \( w \in (0, \infty) \), scale, radius \(a = c - w\), the left endpoint \(b = c + w \), the right endpoint \( c = 0, \; w = 1 \) \([a, b]\) \(f(x) = \frac{3}{2 w} \left(\frac{x - c}{w}\right)^2, \; x \in [a, b]\) \(\{a, b\}\) \( F(x) = \frac{1}{2}\left[1 + \left(\frac{x - c}{w}\right)^3\right], \; x \in [a, b] \) \(Q(p) = c + w(2 p - 1)^{1/3}, \; p \in [0, 1]\) \(c = \frac{a + b}{2}\) \(\frac{3}{5} w^2 = \frac{3}{5}(b - a)^2\) \(0\) \(-\frac{38}{21}\) \(c = \frac{a + b}{2}\) \( c - \frac{1}{2^{1/3}} w \) \( c + \frac{1}{2^{1/3}} w \) location scale rubinstein1973comparative von Mises distribution circular normal distribution Tikhanov distribution continuous The von Mises distribution is used as an approximation to the wrapped normal distribution \(\mu \in (-\infty, \infty)\), the location parameter \(\beta \in (0, \infty)\), the concentration parameter any interval of length \( 2 I \) \(\frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}\) \(\mu\) \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(\mu\) \(\textrm{var}(x)=1-I_1(\kappa)/I_0(\kappa)\)(circular) \(-\beta \frac{I_1(\beta)}{I_0(\beta)} + \ln[2 \pi I_0(\beta)]\) where \(I_n\) is the modfied Bessel function of order \(n\) \(\mu\) \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function location The von Mises distribution is named for Richard von Mises based on his work in diffusion processes. abramowitz1972handbook Wald distribution inverse Gaussian distribution continuous The Wald distribution governs the time that Brownian Motion with positive drift reaches a fixed positive value. \(\mu \in (0, \infty)\), the mean \(\lambda \in (0, \infty)\), the shape parameter \( \mu = 0, \; \lambda = 1 \) \(x\in(0,\infty)\) \(\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp{\frac{-\lambda(x-\mu)^2}{2\mu^2x}}\) \(\mu\left[\left(1+\frac{9\mu^2}{4\lambda^2}\right)^\frac{1}{2}-\frac{3\mu}{2\lambda}\right]\) \(\Phi\left(\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}-1\right)\right)\)\(+\exp\left(\frac{2\lambda}{\mu}\right)\Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1\right)\right)\) where \(\Phi\left(\right)\) is the standard normal distribution function. \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(M(t) = \exp \left[ \frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2 \mu^2}{\lambda} t} \right)\right], \; t \in (-\infty, \frac{\lambda}{2 \mu^2})\) \(\varphi(t) = \exp \left[ \frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2 \mu^2}{\lambda} i t} \right)\right], \; t \in (-\infty, \infty)\) \(\mu\) \(\frac{\mu^3}{\lambda}\) \(3\left(\frac{\mu}{\lambda}\right)^{1/2}\) \(\frac{15\mu}{\lambda}\) \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function The Wald distribution is named for Abraham Wald. chhikara1988inverse Weibull distribution continuous The Weibull distribution is used to model the failure times. \(k \in (0, \infty)\), the shape parameter \(\lambda \in (0, \infty)\), the scale parameter \( k = 1, \; \lambda = 1 \) \( (0, \infty) \) \(f(x) =\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} \exp\left[-\left(\frac{x}{\lambda}\right)^{k}\right], \; x \in (0, \infty)\) 0 if \( k = 1 \), \(\lambda\left(\frac{k-1}{k}\right)^{\frac{1}{k}}\) if \( k \gt 1 \) \(F(x) = 1 - \exp\left[-\left(\frac{x}{\lambda}\right)^k\right], \; x \in (0, \infty)\) \(Q(p) = \lambda \left[- \ln(1 - p)\right]^{1/k}, \; p \in (0, 1)\) \(M(t) = \sum_{n=0}^\infty \frac{t^n \lambda^n}{n!} \Gamma\left(1 + \frac{n}{k}\right), \; t \in (-\infty, \infty), \; k \in (1, \infty)\) where \(\Gamma\) is the gamma function \(\varphi(t) = \sum_{n=0}^\infty \frac{(i t)^n \lambda^n}{n!} \Gamma\left(1 + \frac{n}{k}\right), \; t \in (-\infty, \infty)\) where \(\Gamma\) is the gamma function. \(\lambda \Gamma(1 + 1/k)\) \(\lambda^2 \Gamma(1 + 2/k) - \mu^2\) where \( \mu \) is the mean \(\frac{\Gamma(1 + 3/k)\lambda^3 - 3\mu \sigma^2 - \mu^3}{\sigma^3}\) where \( \mu \) is the mean and \( \sigma \) is the standard deviation \(\lambda[\ln(2)]^{1/k}\) \(\lambda [\ln(4) - \ln(3)]^{1/k}\) \(\lambda [\ln(4)]^{1/k}\) exponential scale The Weibull distribution is named for Waloddi Weibull who published a paper on the distribution in 1951. The distribution was used earlier by Maurice Frechet. The term Weibull distribution was first used in 1955 in a paper by Julius Lieblein. weibull1951statistical zeta distribution Zipf distribution discrete The zeta distribution models ranks and sizes of certain randomly chosen items. \(s \in [1, \infty)\), shape \( s = 1 \) \(\{1, 2, \ldots\}\) \(f(k) = \frac{1}{\zeta(s) k^s}, \; k \in \{1, 2, \ldots\}\) where \( \zeta \) is the zeta function \(1\) \(F(k) = \frac{H_{k,s}}{\zeta(s)}, \; k \in \{1, 2, \ldots\}\) \(Q(p) = F^{-1}(p), \; p \in [0, 1)\) where \(F\) is the distribution function \(\frac{\zeta(s - 1)}{\zeta(s)}\) for \(s \gt 2\) \(\frac{\zeta(s)\zeta(s - 2) - \zeta(s - 1)^2}{\zeta(s)^2}\) for \(s \gt 3\) \( Q\left(\frac{1}{2}\right) \) where \( Q \) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the quantile function The Zipf distribution is named for the American linguist George Kingsley Zipf, who studied the distribution in the context of the frequency of words. johnson2005univariate location-scale distribution continuous Location scale distributions correspond to linear transformations (with positive slope) of a basic random variable, and often correspond to a change of units in a physical problem. the standard distribution, a continuous distribution with support on an interval \(S_0\) \(\mu \in (-\infty, \infty)\), the location parameter \(\sigma \in (0, \infty)\), the scale parameter \(S = \{\mu + \sigma x: x \in S_0\}\) \(f(x) = \frac{1}{\sigma} f_0\left(\frac{x - \mu}{\sigma}\right), \; x \in S\) where \(f_0\) is the probability density function of the standard distribution \(\mu + \sigma x_0\) where \(x_0\) is a mode of the standard distribution \(F(x) = F_0\left(\frac{x - \mu}{\sigma}\right), \; x \in S\) where \(F_0\) is the distribution function of the standard distribution \(Q(p) = \mu + \sigma Q_0(p), \; p \in (0, 1)\) where \(Q_0\) is the quantile function of the standard distribution \(M(t) = e^{\mu t} M_0(\sigma t)\) where \(M_0\) is the moment generating function of the standard distribution \(\varphi(t) = e^{i \mu t} \varphi_0(\sigma t)\) where \(\phi\) is the characteristic function of the standard distribution \(m(n) = \sum_{i=0}^n {n \choose i} \sigma^i \mu^{n-i} m_0(i), \; n \in \{1, 2, \ldots\}\) where \(m_0(i)\) is the \(i\)th raw moment of the standard distribution \(\mu + \sigma \mu_0\) where \(\mu_0\) is the mean of the standard distribution \(\sigma^2 \sigma_0^2\) where \(\sigma_0^2\) is the variance of the standard distribution \(\gamma_{0,1}\) where \(\gamma_{0,1}\) is the skewness of the standard distribution \(\gamma_{0,2}\) where \(\gamma_{0,2}\) is the kurtosis of the standard distribution \(\ln(\sigma) + I_0\) where \(I_0\) is the entropy of the standard distribution \(\mu + \sigma q_{0,2}\) where \(q_{0,2}\) is the median of the standard distribution \(\mu + \sigma q_{0,1}\) where \(q_{0,1}\) is the first quartile of the standard distribution \(\mu + \sigma q_{0,3}\) where \(q_{0,3}\) is the third quartile of the standard distribution meyer1987two folded normal distribution continuous The folded normal distribution governs \(\left|X\right|\) when \(X\) has a normal distribution \(\mu \in (-\infty, \infty)\), the normal mean \(\sigma \in (0, \infty\), the normal standard deviation \( \mu = 0, \; \sigma = 1 \) \([0, \infty)\) \(f(x) = \frac{1}{\sigma \sqrt{2\pi}} \, \exp \left( -\frac{(-x - \mu)^2}{2\sigma^2} \right) + \frac{1}{\sigma \sqrt{2 \pi}} \, \exp \left(-\frac{(x - \mu)^2}{2 \sigma^2} \right), \; x \in [0, \infty) \) \( F(x) = \Phi\left(\frac{x - \mu}{\sigma}\right) + \Phi\left(\frac{x + \mu}{\sigma}\right) - 1, \; x \in [0, \infty) \) where \( \Phi \) is the standard normal distribution function \(F^{-1}(p), p \in (0, 1)\) where \(F\) is the distribution funciton \(\sigma \sqrt{\frac{2}{\pi}} \exp\left(-\frac{\mu^2}{2 \sigma^2} \right) + \mu \left[1 - 2 \Phi\left(-\frac{\mu}{\sigma}\right) \right]\) where \(\Phi\) is the standard normal distribution function \( \mu^2 + \sigma^2 - \left\{ \sigma \sqrt{2/\pi} \exp(-\mu^2 / 2 \sigma^2) + \mu\left[1 - 2 \Phi(-\mu / \sigma)\right] \right\}^2 \) where \( \Phi \) is the standard normal distribution function \(Q(\frac{1}{2})\) where \(Q\) is the quantile function \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the qunatile function leone1961folded half normal distribution continuous The half normal distribution governs \(|X|\) when \(X\) has a normal disstribution with mean 0. \(\sigma \in (0, \infty) \), the scale parameter \( \sigma = 1 \) \([0, \infty)\) \(f(x) = \frac{\sqrt{2}}{\sigma \sqrt{\pi}} \exp \left(-\frac{x^2}{2 \sigma^2} \right), \; x \in (0, \infty)\) 0 \(\sigma \sqrt{\frac{2}{\pi}}\) \( \sigma^2 \left(1 - \frac{2}{\pi}\right) \) \( \frac{1}{2} \ln \left( \frac{ \pi \sigma^2 }{2} \right) + \frac{1}{2}\) \(\mu(n) = \frac{\pi^{(n - 1) / 2}}{\sigma^n} \Gamma\left(\frac{1}{2}(n + 1) \right)\) where \(\Gamma\) is the gamma function \(\frac{\sqrt{2}(4 - \pi)}{(\pi - 2)^{3/2}}\) \(\frac{8(\pi - 3)}{(\pi - 2)^2}\) scale pescim2010beta birthday distribution occupancy distribution discrete This distribution models the number of empty cells when \(n\) balls are distributed at random into \(m\) cells \(m \in \{1, 2, \ldots\}\), the number of cells \(n \in \{1, 2, \ldots\}\), the number of balls \(\{\max\{m-n, 0\}, \ldots, m - 1\}\) \(f(x) = \binom{m}{x} \sum_{j=0}^{m-x} (-1)^j \binom{m - x}{j} \left(1 - \frac{x + j}{m}\right)^n, \quad x \in \{\max\{m-n,0\}, \ldots, m-1\}\) \(F(x) = \sum_{j = 0}^x f(j), \quad x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function \(\mu_{(k)} = \frac{m!}{(m - k)!} \left(\frac{m - k}{m} \right)^n, \quad k \in \{1, 2, \ldots\}\) \(G(t) = \sum_{k=0}^m \binom{m}{k} \left(\frac{m - k}{m}\right)^n (t - 1)^k, \quad t \in R\) \(m \left(1 - \frac{1}{m}\right)^n\) \(m (m - 1) \left(1 - \frac{2}{m}\right)^n + m \left(1 - \frac{1}{m}\right)^n - m^2 \left(1 - \frac{1}{m}\right)^{2n}\) \(\frac{\mu_3 - 3 \mu_1 \mu_2 + 2 \mu_1^2}{\sigma^3}\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation \(\frac{\mu_4 - 4 \mu_1 \mu_3 + 6 \mu_1^2 -3 \mu_1^4}{\sigma^4} - 3\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation \(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function \(Q(\frac{1}{2})\) where \(Q\) is the quantile function \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the qunatile function \(H = -\sum_{x=0}^n \log[f(x)] f(x)\) where \(f\) is the probability density function munford1977note matching distribution discrete The matching distribution governs the number of matches in a random permutation of \(\{1, 2, \ldots, n\}\) \(n \in \{2, 3, \ldots\}\), the number of objects permuted \(\{0, 1, \ldots, n\}\) \(f(x) = \frac{1}{x!} \sum_{j=0}^{n - x} \frac{(-1)^j}{j!}, \; x \in \{0, 1, \ldots, n\}\) \(0\) if \(n\) is even; \(1\) if \(n\) is odd \(F(x) = \sum_{j = 0}^x f(j), \; x \in \{0, 1, \ldots, n\}\) where \(f\) is the probability density function \(\mu_{(k)} = 1, \; k \in \{1, 2, \ldots, n\}; \quad \mu_{(k)} = 0, \; k \in \{n + 1, n + 2, \ldots\}\) \(G(t) = \sum_{k=1}^n \frac{(t-1)^k}{k!}, \; t \in R\) \(1\) \(1\) \(\frac{\mu_3 - 3 \mu_1 \mu_2 + 2 \mu_1^2}{\sigma^3}\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation \(\frac{\mu_4 - 4 \mu_1 \mu_3 + 6 \mu_1^2 -3 \mu_1^4}{\sigma^4} - 3\) where \(\mu_i\) is the \(i\)th raw moment and \(\sigma\) is the standard deviation \(Q(p) = F^{-1}(p), \quad p \in (0, 1)\) where \(F\) is the distribution function \(Q\left(\frac{1}{2}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{1}{4}\right)\) where \(Q\) is the quantile function \(Q\left(\frac{3}{4}\right)\) where \(Q\) is the qunatile function \(H = -\sum_{x=0}^n \log[f(x)] f(x)\) where \(f\) is the probability density function The matching problem was first formulated by Pierre-Redmond Montmort. coupon-collector distribution discrete This distribution models the number number of samples needed to obtain \(k\) distinct values when sampling at random, with replacement from a population of \(m\) objects \(m \in \{1, 2, \ldots\}\), the population size \(k \in \{1, 2, \ldots\}\), the number of distinct values to be obtained \(\{k, k + 1, \ldots\}\) \(f(x) = \binom{m - 1}{k - 1} \sum_{j=0}^{k-1} \binom{k-1}{j} \left(\frac{k - j - 1}{m}\right)^{x-1}, \quad x \in \{k, k + 1, \ldots\}\) \(F(x) = \sum_{j = 0}^x f(j), \quad x \in \{k, k + 1, \ldots\}\) where \(f\) is the probability density function \(G(t) = \prod_{i=1}^k \frac{m - (i - 1)}{m - (i - 1)t}, \quad |t| \lt \frac{m}{k - 1} \) \(\sum_{i=1}^k \frac{m}{m - i + 1}\) \(\sum_{i=1}^k \frac{(i-1)m}{(m - i + 1)^2}\) motwani1995randomized finite order statistic distribution discrete This distribution models an order statistic when a sample is chosen at random, without replacement, from a finite, ordered population \(m \in \{1, 2, \ldots\}\), the population size \(n \in \{1, 2, \ldots, m\}\), the sample size \(k \in \{1, 2, \ldots, n\}\), the the order \(\{k, k + 1, \ldots, m - n + 1\}\) \(f(x) = \frac{\binom{x-1}{k-1} \binom{m-x}{n-k}}{\binom{m}{n}}, \quad x \in \{k, k + 1, \quad m - n + 1\}\) \(k \frac{m+1}{n+1}\) \(k(n - k + 1) \frac{(m + 1)(m - n)}{(m + 1)^2 (n + 2)}\) Erlang distribution continuous The Erlang probability distribution is related exponential and Gamma distributions and is used to examine the number of event arrivals. For instance telephone calls which might be made at the same time to the operators of the switching stations. This work on telephone traffic engineering has been expanded to consider waiting times in queueing systems in general \(k \in \mathbb{N}\), shape parameter \(\lambda > 0\), rate parameter \(\theta = 1/\lambda > 0\), scale parameter \(\scriptstyle x\;\in\;[0,\,\infty)\!\) \(\scriptstyle\frac{\lambda^k x^{k-1}e^{-\lambda x}}{(k-1)!\,}\) \(\scriptstyle\frac{\gamma(k,\,\lambda x)}{(k\,-\,1)!}\;=\;1\,-\,\sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}\) \(\scriptstyle\frac{k}{\lambda}\,\) No simple closed form \(\scriptstyle\frac{k}{\lambda^2}\,\) angusintroduction Generalized Gamma distribution continuous The generalized gamma distribution is not often used to model life data by itself, but it is sometimes used to determine which of those life distributions should be used to model a particular set of data. \(\alpha \in (0, \infty)\), the scale parameter \(\beta \in (0, \infty)\), the shape parameter \(\gamma \in (0, \infty)\), the shape parameter \((0, \infty)\) \(f(x; a, d, p) = \frac{(p/a^d) x^{d-1} e^{-(x/a)^p}}{\Gamma(d/p)}\) \(F(x; a, d, p) = \frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)}\) where \(\gamma\) denotes incomplete gamma function stacy1962generalization Makeham distribution Gompertz–Makeham law of mortality continuous The Gompertz–Makeham law states that the death rate is the sum of an age-independent component and an age-dependent component, which increases exponentially with age. \(\gamma \in (0, \infty)\) \(\delta \in (0, \infty)\) \(\kappa \in (0, \infty)\) \((0, \infty)\) \(f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta(\kappa^x-1)}{log(\kappa)})\) \(1-\exp(-\lambda x-\frac{\alpha}{\beta}(e^{\beta x}-1))\) gavrilov1983human hypoexponential distribution generalized Erlang distribution continuous The hypoexponential distribution has a coefficient variation less than one, compared to the hyperexponential distribution which has a coefficient variation greater than one and the exponential distribution which has a coefficient variation of one. \(\alpha_1,...,\alpha_n \in (0, \infty), \alpha_i \neq \alpha_j for i \neq j\) \([0, \infty)\) \(f(x) = \sum_{i=1}^{n}(\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j})\) Expressed as a phase-type distribution: \(1-\boldsymbol{\alpha}e^{x\Theta}\boldsymbol{1}\) \(\boldsymbol{\alpha}(tI-\Theta)^{-1}\Theta\mathbf{1}\) \(\sum^{k}_{i=1}1/\lambda_{i}\,\) \((k-1)/\lambda\) if \(\lambda_{k} = \lambda\) \(\sum^{k}_{i=1}1/\lambda^2_{i}\) \(\ln(2)\sum^{k}_{i=1}1/\lambda_{i}\) \(2(\sum^{k}_{i=1}1/\lambda_{i}^3)/(\sum^{k}_{i=1}1/\lambda_{i}^2)^{3/2}\) bolch2006queueing doubly noncentral t distribution continuous The doubly noncentral t distribution is an extended version of the singly noncentral t distribution in that it has two noncentrality parameters instead of just one. See http://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.1969.tb00102.x/pdf krishnan1968series hyperexponential distribution continuous The hyperexponential distribution has a coefficient variation greater than one, compared to the hypoexponential distribution which has a coefficient variation less than one and the exponential distribution which has a coefficient variation of one. \(\alpha_1,...,\alpha_n \in (0, \infty), \alpha_i \neq \alpha_j for i \neq j\) \(p_i > 0, \sum_{i=1}^{n} p_i = 1\) \((0, \infty)\) \(f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}\) singh2007estimation Muth distribution continuous The Muth distribution is related to reliability models. It has mostly a theoretical interest. Muth distribution has two basic properties: (i) the mode of this random model is a function involving the golden ratio and (ii) the second non-central moment can be expressed in terms of the exponential integral function. The moments of higher order cannot be expressed in a simple way. The Muth distribution does not have the variate generation property for simulation purposes. Its quantile function can be expressed in closed form in terms of the negative branch of the Lambert W function. The limit distributions of the maxima and minima of the Muth distribution are the Gumbel and Weibull distributions, respectively. \(\kappa \in [0, 1]\), the shape parameter \((0, \infty)\) \(f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}\) muth1960optimal generalized error distribution generalized normal distribution generalized Gaussian distribution exponential power distribution continuous The error distribution is a parametric family of symmetric distributions. It adds a shape parameter to the normal distribution. \(a \in (-\infty, \infty)\), the mean \(b \in (0, \infty)\), the scale parameter \(c \in (0, \infty)\), the shape parameter \((-\infty, \infty)\) \(f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}\) hosking2005regional minimax distribution continuous The Minimax distribution is an alternative two-parameter distribution of the Beta distribution. \(\beta \in (0, \infty)\) \(\gamma \in (0, \infty)\) \((0, 1)\) \(f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}\) marchand2002minimax noncentral F distribution continuous The noncentral F distribution is a generalization of the ordinary F distribution. \(\delta \in (0, \infty)\) \((0, \infty)\) \(f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}\) increasing-decreasing-bathtub distribution continuous The IDB distribution can be used to model either an increasing, decreasing or bathtub shaped failure rate function, which is a combination of a linearly increasing failure rate and a decreasing failure rate function \(\delta \in (0, \infty)\) \(\kappa \in (0, \infty)\) \(\gamma \in [0, \infty)\) \((0, \infty)\) \(f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}\) \(F(x|d_1,d_2,\lambda)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\frac{\lambda}{2}}\right)I\left(\frac{d_1F}{d_2 + d_1F}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right)\) where \(I\) is the regularized incomplete beta function 2\\ \mbox{Does not exist} &\nu_2\le2\\ \end{cases}\) ]]> 4\\ \mbox{Does not exist} &\nu_2\le4.\\ \end{cases}\) ]]> kay1993fundamentals Benford's law first digit law significant digit law discrete Benford's law states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. \(b \in \{2, 3, \ldots\}\), the base \(b = 10\) \(\{1, 2, \ldots, b - 1\}\) \(f(x) = \log_b(x + 1)- \log_b(x) = \log_b\left(\frac{x + 1}{x}\right), \; x \in \{1, 2, \ldots, b - 1\}\) hill1995statistical doubly noncentral F distribution continuous The doubly noncentral F distribution is an extended version of the singly noncentral F distribution in that it has two noncentrality parameters instead of just one. \(\delta \in (0, \infty)\) \(\gamma \in (0, \infty)\) \((0, \infty)\) \(f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}\) bulgren1971representations two-sided power distribution continuous The two-sided power distribution is an alternative to the triangular distribution, allowing for a nonlinear distribution. Triangular and uniform distributions are special cases of the two-sided power distribution. \(n \in (0, \infty)\) \(a \in (-\infty, \infty)\) \(b \in (a, \infty)\) \(m=(b-a)\theta+a\) \((a, b)\) \(f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a\lt x\le m \\ \frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x\lt b \end{cases}\) van2002standard extreme value distribution Gumbel distribution continuous The Extreme Value distribution is the limiting distribution of the minimum of a large number of unbounded identically distributed random variables. \(a \in (-\infty, \infty)\), location \(b \in (0, \infty)\), scale \( a = 0, \; b = 1 \) \((-\infty, \infty)\) \( f(x) = \frac{1}{b} \exp\left(-\frac{x - a}{b}\right) \exp\left[-\exp\left(-\frac{x - a}{b}\right)\right], \; x \in (-\infty, \infty) \) \( a \) \( F(x) = \exp\left[-\exp\left(-\frac{x - a}{b}\right)\right], \; x \in (-\infty, \infty) \) \( F^{-1}(p) = a - b \ln[-\ln(p)], \; p \in (0, 1) \) \( M(t) = e^{a t} \Gamma(1 - b t), \; t \in (-\infty, 1/b) \) where \( \Gamma \) is the gamma function \( a + b \gamma \) where \( \gamma \) is Euler's constant \( b^2 \pi / 6 \) \( 12 \sqrt{6} \zeta(3) / \pi^3 \) \( 12/5 \) \( a - b \ln[\ln(2)] \) \(a - b \ln[\ln(4) - \ln(3)]\) \(a - b \ln[\ln(4)]\) \( \ln(b) + \gamma + 1 \) location scale The Gumbel distribution is named for Emil Gumbel, who derived it in his study of extreme values in 1954. embrechts2011modelling Lomax distribution Pareto Type II distribution continuous The Lomax distribution is essentially a Pareto distribution that has been shifted so that its support begins at zero. \(\kappa \in (0, \infty)\), the shape parameter \(\lambda \in (0, \infty)\), the scale parameter \(x\ge0\) \({\alpha\over\lambda}\left[{1+{x\over\lambda}}\right]^{-(\alpha+1)}\) \(e^{-t(x)},\,\) \(\log(\sigma)\,+\,\gamma\xi\,+\,(\gamma+1)\) coles2001introduction generalized Pareto distribution continuous The generalized Pareto distribution allows a continuous range of possible shapes that includes both the exponential and Pareto distributions as special cases. \(\kappa \in (-\infty, \infty)\), the shape parameter \(\sigma \in (0, \infty)\), the scale parameter \(\mu \in (0, \infty)\), the mean \(\mu\leqslant x\leqslant\mu-\sigma/\xi\,\;(\xi<0)\) ]]> \(\frac{1}{\sigma}(1+\xi z)^{-(1/\xi+1)}\)
where\(z=\frac{x-\mu}{\sigma}\) \(1-(1+\xi z)^{-1/\xi}\,\) \(\mu+\frac{\sigma(2^{\xi}-1)}{\xi}\) chotikapanich2008modeling Kolmogorov-Smirnov test Kolmogorov distribution continuous The Kolmogorov-Smirnov test can be modified to serve as a goodness of fit test. none \((0, \infty)\) \(f(x)=1-2[\exp{-x^2}-\exp{-4 x^2}+\exp{-9 x^2}-\exp{-16 x^2}+...]\) kolmogorov1933sulla logistic-exponential distribution continuous infinte \( \alpha = \beta = \frac{1}{2} \) lan2008logistic power-function distribution continuous nonsymmetric finite \( f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} \) moothathu1986characterization student t non-central distribution continuous nonsymmetric infinite \( f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx \) 2 ;\\ \mbox{Does not exist}, &\mbox{if }\nu\le2 .\\ \end{cases}\) ]]> \(\sqrt{\frac{\nu}{2}}\frac{\Gamma\left(\frac{\nu+2}{2}\right)}{\Gamma\left(\frac{\nu+3}{2}\right)}\mu;\,\) lenth1989algorithm inverted gamma distribution continuous nonsymmetric finite positive \( \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right) \) \(\frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!\) \(\frac{2\left(-\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4\beta t}\right)\) \(\frac{\beta}{\alpha-1}\!\) for \(\alpha > 1\) \(\frac{\beta}{\alpha+1}\!\) \(\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\!\) for \(\alpha > 2\) \(\frac{4\sqrt{\alpha-2}}{\alpha-3}\!\) for \(\alpha > 3\) \(\frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\!\) for \(\alpha > 4\) \(\alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\Psi(\alpha)\) witkovsky2001computing Fisher-Tippett distribution continuous nonsymmetric finite positive \( \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right) \) embrechts2011modelling Gibrat's distribution continuous nonsymmetric finite positive \( \frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right] \) eeckhout2004gibrat Gompertz distribution continuous nonsymmetric finite positive \( b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right] \) \(1-\exp\left(-\eta\left(e^{bx}-1 \right)\right)\) \(\text{E}\left(e^{-t x}\right)=\eta e^{\eta}\text{E}_{t/b}\left(\eta\right)\), \(\text{with E}_{t/b}\left(\eta\right)=\int_1^\infty e^{-\eta v} v^{-t/b}dv,\ t>0\) \((-1/b)e^{\eta}\text{Ei}\left(-\eta\right)\), \(\text {where Ei}\left(z\right)=\int\limits_{-z}^{\infty}\left(e^{-v}/v\right)dv\) \(\left(1/b\right)\ln\left[\left(-1/\eta\right)\ln\left(1/2\right)+1\right]\) bemmaor2012modeling multinomial distribution discrete nonsymmetric finite positive \(X_i \in \{0,\dots,n\}\), where \(\Sigma X_i = n\!\) \( f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k} \) \(\biggl( \sum_{i=1}^k p_i e^{t_i} \biggr)^n\) \(E\{X_i\} = np_i\) \(\textstyle{\mathrm{Var}}(X_i) = n p_i (1-p_i)\), where \(\textstyle {\mathrm{Cov}}(X_i,X_j) = - n p_i p_j~~(i\neq j)\) evans2000statistical negative multinomial distribution discrete nonsymmetric finite positive \( f(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} \) \(\tfrac{k_0}{p_0}\,p\) \(\tfrac{k_0}{p_0^2}\,pp' + \tfrac{k_0}{p_0}\,\operatorname{diag}(p)\) gall2006modes negative hypergeometric distribution discrete finite positive If \({x \choose 2} \lt\lt W\) and \({b \choose 2} \lt\lt B\) then \(X\) can be approximated as a negative binomial random variable with parameters \(r = b\) and \(p = \frac{W}{W+B}\). This approximation simplifies the distribution by looking as a system with replacement for large values of \(W\) and \(B\). \(W \in \{1,2,...\}\) \(B \in \{1,2,...\}\) \(b \in \{1,2,...,B\}\) \(x=\{0,1,...,W\}\) \( f(x) \frac{ { x+b-1 \choose x} {W+B-b-x \choose W-x} }{ {W+B \choose W} } \) askey2010generalized power series distribution discrete infinite positive \( f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \! \) yanushauskas1980double beta-Pascal distribution discrete infinite positive \( f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \! \) johnson2005univariate gamma-Poisson distribution discrete infinite positive \(\{k, k+1, \ldots\}\) \(f(x) = {x-1 \choose k-1} p^x (1 - p)^{x-k}, \; x \in \{k, k+1, \ldots\}\) \(\lfloor 1 + \frac{k-1}{p}\rfloor\) \(F(x) = \sum_{j=k}^x f(j) , \; x \in \{k, k+1, \ldots\}\) where \(f\) is the probability density function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(G(t) = \left[\frac{p t}{1 - (1-p) t}\right]^k, \; t \in (-\frac{1}{1-p}, \frac{1}{1-p})\) \(M(t) = \left[\frac{p e^t}{1 - (1-p) e^t}\right]^k, \; t \in (-\infty, -\ln(1 - p))\) \(\varphi(t) = \left[\frac{p e^{i t}}{1 - (1-p) e^{i t}}\right]^k, \; t\in (-\infty, \infty)\) \(k \frac{1}{p}\) \(k \frac{1-p}{p^2}\) \(\frac{2-p}{\sqrt{k (1-p)}}\) \(\frac{1}{k} \left[6 + \frac{p^2}{1 - p}\right]\) \(Q(\frac{1}{2})\) where \(Q\) is the quantile function \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function The alternative name Pascal distribution is in honor of Blaise Pascal who used the distribution in his solution to the Problem of Points. el2006negative Polya distribution discrete finite positive \(\{k, k+1, \ldots\}\) \(f(x) = {x-1 \choose k-1} p^x (1 - p)^{x-k}, \; x \in \{k, k+1, \ldots\}\) \(\lfloor 1 + \frac{k-1}{p}\rfloor\) \(F(x) = \sum_{j=k}^x f(j) , \; x \in \{k, k+1, \ldots\}\) where \(f\) is the probability density function \(Q(p) = F^{-1}(p), \; p \in (0, 1)\) where \(F\) is the distribution function \(G(t) = \left[\frac{p t}{1 - (1-p) t}\right]^k, \; t \in (-\frac{1}{1-p}, \frac{1}{1-p})\) \(M(t) = \left[\frac{p e^t}{1 - (1-p) e^t}\right]^k, \; t \in (-\infty, -\ln(1 - p))\) \(\varphi(t) = \left[\frac{p e^{i t}}{1 - (1-p) e^{i t}}\right]^k, \; t\in (-\infty, \infty)\) \(k \frac{1}{p}\) \(k \frac{1-p}{p^2}\) \(\frac{2-p}{\sqrt{k (1-p)}}\) \(\frac{1}{k} \left[6 + \frac{p^2}{1 - p}\right]\) \(Q(\frac{1}{2})\) where \(Q\) is the quantile function \(Q(\frac{1}{4})\) where \(Q\) is the quantile function \(Q(\frac{3}{4})\) where \(Q\) is the quantile function The alternative name Pascal distribution is in honor of Blaise Pascal who used the distribution in his solution to the Problem of Points. el2006negative gamma-normal distribution bivariate continuous infinite \( f(x,\tau|\mu,\lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt{\lambda}}{\Gamma(\alpha)\sqrt{2\pi}} \, \tau^{\alpha-\frac{1}{2}}\,e^{-\beta\tau}\,e^{ -\frac{ \lambda \tau (x- \mu)^2}{2}} \) \(\operatorname{E}(X)=\mu\,\! ,\quad \operatorname{E}(\tau)= \alpha \beta^{-1}\) \(\operatorname{var}(X)= \frac{\beta}{\lambda (\alpha-1)} ,\quad \operatorname{var}(\tau)=\alpha \beta^{-2}\) bernardo2001bayesian discrete Weibull distribution discrete infinite positive \( f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \! \) englehardt2012methods noncentral beta distribution continuous positive \( f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1) \! \) \(F(x) = \sum_{j=0}^\infty \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}I_x(a+j,b)\) where \(I_x\) is regularized incomplete beta function abramowitz1972handbook arctangent distribution continuous infinite positive \( f(x; \lambda, \phi)= \frac{\lambda}{[\arctan(\lambda \phi)+\pi/2] [1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty \lt \lambda \lt \infty) \! \) pollastri2004some log-gamma distribution continuous infinite \( f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}\), where \((-\infty \lt x \lt \infty) \! \) demirhan2011multivariate Bernoulli distribution binomial distribution If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent Bernoulli variables, each with parameter \(p \in [0, 1]\) then \(Y = \sum_{i = 1}^n X_i\) has the binomial distribution with parameters \(n\) and \(p\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Bernoulli distribution geometric distribution If \((X_1, X_2, \ldots)\) is a sequence of independent Bernoulli variables, each with parameter \(p \in (0, 1)\), then \(Y = \min\{n \in \{1, 2, \ldots\}: X_n = 1\}\) has the geometric distribution with parameter \(p\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Bernoulli distribution negative binomial distribution If \((X_1, X_2, \ldots)\) is a sequence of independent Bernoulli variables, each with parameter \(p \in (0, 1)\), then for \(ki \in \{1, 2, \ldots\}\), \(Y = \min\{n \in \{1, 2, \ldots\}: \sum_{i=1}^n X_i = k\}\) has the negative binomial distribution with parmeters \(k\) and \(p\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Bernoulli distribution Rademacher distribution If \(X\) has the Bernoulli distribution with parameter \(\frac{1}{2}\) then \(2 X - 1\) has the Rademacher distribution. linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution arcsine distribution The beta distribution with parameters \(\alpha = \frac{1}{2}\) and \(\beta = \frac{1}{2}\) is the arcsine distribution. special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution continuous uniform distribution The beta distribution with parameters \(\alpha = 1\) and \(\beta = 1\) is the (standard) continuous uniform distribution. Also, the k-th order statistic from sample of n independent continuous uniform variables is Beta(k,n+1-k). special case convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta general distribution beta distribution If \(X\) has the beta general distribution with parameters \(\alpha \in (0, \infty)\), \(\beta \in (0, \infty)\) and \(L \lt R\), then \(Y = \frac{X-L}{R-L}\) has beta distribution with parameters \(\alpha\) and \(\beta\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution inverse beta distribution If \(X\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\), then \(Y = \frac{X}{1 - X}\) has the inverse beta distribution with parameters \(\alpha\) and \(\beta\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution semicircle distribution If \(X\) has the beta distribution with parameters \(\alpha = \frac{3}{2}\) and \(\beta = \frac{3}{2}\), and \(r \in (0, \infty)\), then \(Y = r (2 X - 1)\) has the semicircle distribution with parameter \(r\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution beta distribution If \(X\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) then \(Y = 1 - X\) has the beta distribution with parameters \(\beta\) and \(\alpha\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution beta distribution If \(X\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta = 1\), and \(r \in (0, \infty)\), then \(Y = X^r\) has the beta distribution with parameters \(\frac{\alpha}{r}\) and \(1\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution Pareto distribution If \(X\) has the beta distribution with left parameter \(\alpha \in (0, \infty)\) and right parameter \(1\), then \(Y = \frac{1}{X}\) has the Pareto distribution with shape parameter \(\alpha\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution binomial distribution beta-binomial distribution If \(P\) has the beta distribution with parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) and if the conditional distribution of \(X\) given \(P = p\) has the binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and \(p\), then \(X\) has the beta-binomial distribution with parameters \(n\), \(\alpha\), and \(\beta\). Conditioning doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta-binomial distribution continuous uniform distribution The standard uniform distribution is a special case of beta-binomial distribution with \( \a=\b=1 \) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta-binomial distribution negative hypergeometric distribution The negative hypergeometric distribution is a special case of beta-binomial distribution with \( n=n_1, a=n_2, b=n_3 \) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 binomial distribution standard normal distribution If \(X_n\) has the binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and fixed \(p \in (0, 1)\) then then the distribution of \(Z_n = \frac{X_n - n p}{\sqrt{n p (1 - p)}}\) converges to the standard normal distribution as \(n \to \infty\). central limit theorem dinov2008central doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 binomial distribution Bernoulli distribution The binomial distribution with parameters \(n = 1\) and \(p \in [0, 1]\) is the Bernoulli distribution with parameter \(p\). Also, if \((X_1, X_2, \ldots, X_n)\) is a sequence of independent Bernoulli variables, each with parameter \(p \in [0, 1]\) then \(Y = \sum_{i = 1}^n X_i\) has the binomial distribution with parameters \(n\) and \(p\). special case convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 binomial distribution binomial distribution If \(X\) has the binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and \(p \in [0, 1]\); \(Y\) has the binomial distribution with parameters \(m \in \{1, 2, \ldots\}\) and \(p\); and \(X\) and \(Y\) are independent, then \(X + Y\) has the binomial distribution with parameters \(m + n\) and \(p\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 binomial distribution hypergeometric distribution Suppose that \(\boldsymbol{X} = (X_1, X_2, \ldots)\) is a Bernoulli trials sequence with parameter \(p \in (0, 1)\). For \(n \in \{1, 2, \ldots\}\) let \(Y_n = \sum_{i=1}^n X_i\), so that \(Y_n\) has the binomial distribution with parameters \(n\) and \(p\). If \(m \lt n\) then the distribution of \(Y_m\) given \(Y_n = k\) is hypergeoemtric with parameters \(m\), \(n\), and \(k\). Conditional distribution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 binomial distribution Poisson distribution The binomial distribution with parameters \(n \in \{1, 2, \ldots\}\) and \(p \in (0, 1)\) converges to the Poisson distribution with parameter \(\lambda \in (0, \infty)\) if \(n \to \infty\), \(p \to 0\), with \(n p \to \lambda\). parameter limit doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 binomial distribution negative binomial distribution For \(n \in \{1, 2, \ldots\}\), let \(Y_n\) denote the number of successes in the first \(n\) of a sequence of Bernoulli trials, so that \(Y_n\) has the binomial distribution with trial parameter \(n\) and sucess parameter \(p\). Then for \(k \in \{1, 2, \ldots\}\), \(Z_k = \min\{n: Y_n \geq k\} - k\) has the negative binomial distribution with stopping parameter \(k\) and success parameter \(p\). inverse stochastic process doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Cauchy Distribution Cauchy Distribution If \(X\) has the Cauchy distribution with location parameter \(\alpha_1 \in (-\infty, \infty)\) and location parameter \(\beta_1 \in (0, \infty)\), \(Y\) has the Cauchy distribution with location parameter \(\alpha_2 \in (-\infty, \infty)\) and scale parameter \(\beta_2 \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the Cauchy distribution with location parameter \(\alpha_1 + \alpha_2\) and scale parameter \(\beta_1 + \beta_2\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Cauchy Distribution normal Distribution If \(Z_1\) and \(Z_2\) are two independent standard normal variables, then the ratio \(\frac{Z_1}{Z_2}\) has the standard Cauchy distribution. transformation MR1326603 Cauchy Distribution normal Distribution If \(Z_1\) and \(Z_2\) are two independent normally distributed random variables \(\sim N(0, \sigma^2)\), then the ratio \(\frac{Z_1}{Z_2}\) has the standard Cauchy distribution. transformation MR1326603 Cauchy distribution Cauchy distribution If \(X\) has Cauchy distribution with location parameter \(\alpha \in (-\infty, \infty)\) and scale parameter \(\beta \in (0, \infty)\), \(a \in (-\infty, \infty)\) and \(b \in (0, \infty)\), then \(a + b X\) has the Cauchy distribution with location parameter \(a + b \alpha\) and location parameter \(\beta b\). location-scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution chi-square distribution If \(X\) has the chi-square distribution with \(m \in (0, \infty)\) degrees of freedom; \(Y\) has the chi-square distribution with \(n \in (0, \infty)\) degrees of freedom; and \(X\) and \(Y\) are independent, then \(X + Y\) has the chi-square distribution with \(m + n\) degrees of freedom. convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution gamma distribution If \(X\) has a chi-square distribution with \(\nu \in \{1, 2, \ldots\}\) degrees of freedom, and \(c \in (0, \infty)\) , then \(Y = c X\) has the gamma distribution with shape parameter \(k = \frac{\nu}{2}\) and scale parameter \(\theta = 2 c\). scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution normal distribution If \(X_n\) has the chi-square distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom, then the distribution of \(Z = \frac{X_n - n}{\sqrt{2 n}}\) converges to the standard normal distribution as \(n \to \infty\). central limit theorem doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 dinov2008central chi-square distribution F-distribution If \(U\) has the chi-square distribution with \(m \in \{1, 2, \ldots\}\) degrees of freedom; \(V\) has the chi-square distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom; and \(U\) and \(V\) are independent, then \(X = \frac{U/m}{V/n}\) hs the \(F\)-distribution with \(m\) degrees of freedom in the numerator and \(n\) degrees of freedom in the denominator nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 non-central chi-square distribution chi-square distribution If \(X\) has the non-central chi-square distribution with \(\nu \in \{1, 2, \ldots\}\) degrees of freedom and non-centrality parameter \(\lambda = 0\), then \(X\) has a chi-square distribution with \(\nu\) degrees of freedom. special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution chi distribution If \(X\) has the chi-square distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom, then \(\sqrt{X}\) has the chi distribution with \(n\) degrees of freedom. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution normal distribution Students t distribution If \(Z\) has the standard normal distribution, \(V\) has the chi-square distribution with \(n \in (0, \infty)\) degrees of freedom, and \(Z\) and \(V\) are independent, then \(T = \frac{Z}{\sqrt{V / n}}\) has the students's \(t\)-distribution with \(n\) degrees of freedom. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution Poisson distribution Rice distribution If \(X\) has the Poisson distribution with parameter \(\frac{\nu^2}{2 \sigma^2}\) where \(\nu \in (0, \infty)\) and \(\sigma \in (0, \infty)\), and the conditional distribution of \(Y\) given \(X = x \in \{0, 1, 2, \ldots\}\) is chi-square with \(2 x + 2\) degrees of freedom, then \(\sigma \sqrt{X}\) has the Rice distribution with distance parameter \(\nu\) and scale parameter \(\sigma\). mixture and transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution continuous uniform distribution If \(X\) is uniformly distributed on the interval \([a, b]\) and \(c, d \in (-\infty, \infty)\) with \(c \ne 0\), then \(Y = cX + d\) is uniformly distributed on \([ca + d, cb + d]\) if \(c \gt 0\) or on \([cb + d, ca + d]\) if \(c \lt 0\) linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution standard uniform distribution The continuous uniform distribution on \([0, 1]\) is the standard uniform distribution special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution triangular distribution If \(X\) and \(Y\) are independent and each is uniformly distributed on the interval \([a, b]\), then \(X + Y\) has the triangular distribution with parameters \(a\), \(b\), and \(c = \frac{a+b}{2}\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution exponential distribution If \(X\) has the standard uniform distribution and \(\beta \in (0, \infty)\), then \(-\beta \ln(1 - X)\) has the exponential distribution with scale parameter \(\beta\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Pareto distribution If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\beta \in (0, \infty)\) then \(\frac{\mu}{(1 - X)^{1/\beta}}\) has the Pareto distribution with location parameter \(\mu\) and shape parameter \(\beta\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution beta distribution If \(X\) has the standard uniform distribution and \(\alpha \in (0, \infty)\) then \(X^{1/\alpha}\) has the beta distribution with left parameter \(\alpha\) and right parameter 1. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Cauchy distribution If \(X\) has the standard uniform distribution then \(\tan[\pi(X - \frac{1}{2})]\) has the standard Cauchy distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution arcsine distribution If \(X\) has the standard uniform distribution then \(\sin^2(\frac{\pi}{2} X)\) has the arcsine distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution exponential-logarithmic distribution If \(X\) has the standard uniform distribution, \(b \in (0, \infty)\), and \(p \in (0, 1)\) then \(\frac{1}{b}\ln\left(\frac{1 - p}{1 - p^{1 - X}}\right)\) has the exponential-logarithmic distribution with parameters \(b\) and \(p\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution geometric distribution If \(X\) has the standard uniform distribution and \(p \in (0, 1)\) then \(\lceil \frac{\ln(1 - X)}{\ln(1 - p)}\rceil\) has the geometric distribution with parameter \(p\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Gumbel distribution If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\sigma \in (0, \infty)\) then \(\mu - \sigma \ln(-\ln(X))\) has the Gumbel distribution with location prarameter \(\mu\) and scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution hyperbolic secant distribution If \(X\) has the standard uniform distribution then \(\frac{2}{\pi} \ln[\tan(\frac{\pi}{2} X)]\) has the hyperbolic secant distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Laplace distribution If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), \(b \in (0, \infty)\), then \(\mu + b \ln(2 \min\{X, 1 - X\})\) has the Laplace distribution with location parameter \(\mu\) and scale parameter \(b\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution logistic distribution If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\sigma \in (0, \infty)\), then \(\mu + \sigma \ln\left(\frac{X}{1 - X}\right)\) has the logistic distribution with location parameter \(\mu\) and scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution log-logistic distribution If \(X\) has the standard uniform distribution, \(\alpha \in (0, \infty)\), and \(\beta \in (0, \infty)\), then \(\alpha \left(\frac{X}{1 - X}\right)^{1/\beta}\) has the log-logistic distribution with scale parameter \(\alpha\) and shape parameter \(\beta\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Rayleigh distribution If \(X\) has the standard uniform distribution and \(\sigma \in (0, \infty)\), then \(\sigma \sqrt{-2 \ln(1 - X)}\) has the Rayleigh distribution with scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Weibull distribution If \(X\) has the standard uniform distribution, \(\sigma \in (0, \infty)\) and \(\alpha \in (0, \infty)\), then \(\sigma (-\ln(1 - X))^{1/\alpha}\) has the Weibull distribution with shape parameter \(\alpha\) and scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Irwin-Hall distribution If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the standard uniform distribution, then \(sum_{i=1}^n X_i\) has the Irwin-Hall distribution with parameter \(n\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution exponential distribution If \(X\) has the exponential distribution with rate parameter \(r \in (0, \infty)\), \(Y\) has the exponential distribution with rate parameter \(s \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(\min\{X, Y\}\) has the exponential distribution with rate parameter \(r + s\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution continuous uniform distribution If \(X\) has the exponential distribution with parameter \(\lambda \in (0, \infty)\) then \(Y = e^{-\lambda X}\) has the standard uniform distribution. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution gamma distribution If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the exponential distribution with parameter \(\lambda \in (0, \infty)\) then \(Y = \sum_{i=1}^n X_i\) has the gamma distribution with shape parameter \(n\) and scale parameter \(\frac{1}{\lambda}\). Also, If \(X\) has the gamma distributed with parameter shape parameter \(k = 1\) and scale parameter \(\lambda \in (0, \infty)\) then and then \(X\) has the exponential distribution with scale parameter \(\lambda\) (and hence rate parameter \(1/\lambda\). convolution special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution Pareto distribution If \(a \in (0, \infty)\) and \(X\) has the exponential distribution with parameter \(\lambda \in (0, \infty)\) then \(Y = a e^X\) has the Pareto distribution with scale parameter \(a\) and shape parameter \(\lambda\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution Weibull distribution If \(X\) has the standard exponential distribution, \(k \in (0, \infty)\), and \(b \in (0, \infty)\), then \(Y = b X^{1/k}\) has the Weibull distribution with shape parameter \(k\) and scale parameter \(b\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution extreme value distribution If \((X_1, X_2, \ldots)\) is a sequence of indpendent random variables, each with the standard exponential distribution, then the distribution of \(\max\{X_1, \ldots, X_n\} - \ln(n)\) converges to the standard Gumbel distribution. limiting distribution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution Laplace distribution If \(X\) and \(Y\) are independent random variables and each has the exponential distribution with scale parameter \(\sigma \in (0, \infty)\) then \(X - Y\) has the Laplace distribution with location parameter \(0\) and scale parameter \(\sigma\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution Rademacher distribution Laplace distribution If \(X\) has the exponential distribution with scale parameter \(\sigma \in (0, \infty)\), \(Y\) has the Rademacher distribution, and \(X\) and \(Y\) are independent, then \(X V\) has the Laplace distribution with location parameter \(0\) and scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution normal distribution Laplace distribution If \(X\) has the standard exponential distribution, \(Z\) has the standard normal distribution, \(X\) and \(Z\) are independent, \(\mu \in (-\infty, \infty)\), and \(\sigma \in (0, \infty)\), then \(\mu + \sigma Z \sqrt{2 X}\) has the Laplace distribution with location parameter \(\mu\) and scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 F-distribution F-distribution If \(X\) has the F-distribution with \(m \in \{1, 2, \ldots\}\) degrees of freedom in the numerator and \(n \in \{1, 2, \ldots\}\) degrees of freedom in the denominator, then \(\frac{1}{X}\) has the F-distribution with \(n\) degrees of freedom in the numerator and \(m\) degrees of freedom in the denominator. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 F-distribution beta distribution If \(X\) has the F-distribution with \(m \in (0, \infty)\) degrees of freedom in the numerator and \(n \in (0, \infty)\) degrees of freedom in the denominator, then \(\frac{(m/n)X}{1 + (m/n)X}\) has the beta distribution with left parameter \(\frac{m}{2}\) and right parameter \(\frac{n}{2}\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 F-distribution chi-square distribution If \(X\) has the chi-square distribution with \(m \in \{1, 2, \ldots\}\) degrees of freedom in the numerator and \(n \in \{1, 2, \ldots\}\) degrees of freedom in the denominator, then the distribution of \(m X\) converges to the chi-square distribution with \(m\) degrees of freedom as \(n \to \infty\). limiting distribution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution gamma distribution If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\) and \(c \in (0, \infty)\), then \(Y = cX\) has the gamma distribution with shape parameter \(\alpha\) and scale parameter \(c \lambda\). scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution gamma distribution If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), \(Y\) has the gamma distribution with shape parameter \(\beta \in (0, \infty)\) and scale parameter \(\lambda\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the gamma distribution with shape parameter \(\alpha + \beta\) and scale parameter \(\lambda\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution exponential distribution If \(X\) has the gamma distributed with parameter shape parameter \(k = 1\) and scale parameter \(\lambda \in (0, \infty)\) then and then \(X\) has the exponential distribution with scale parameter \(\lambda\) (and hence rate parameter \(1/\lambda\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution chi-square distribution If \(X\) has the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), then \(\frac{2 X}{\lambda}\) has the chi-square distribution with \(k\) degrees of freedom. linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution Erlang distribution If \(X\) has the gamma distribution with shape parameter \(k \in \{1, 2, \ldots\}\) and scale parameter \(c \in (0, \infty)\), then \(X\) has the Erlang distribution with shape parameter \(k\) and scale parameter \(c\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution Maxwell-Boltzmann distribution If \(X\) has the gamma distribution with shape parameter \(k = \frac{3}{2}\) and scale parameter \(\theta = 2 a^2\) where \(a \in (0, \infty)\), then \(\sqrt{X}\) has the Maxwell-Boltzmann distribution with parameter \(a\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution normal distribution If \(X_k\) has the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) then the distribution of \(\frac{X - k b}{\sqrt{k} b}\) converges to the standard normal distribution as \(k \to \infty\). central limit theorem doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution beta distribution If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), \(Y\) has the gamma distribution with shape parameter \(\beta \in (0, \infty)\) and scale parameter \(\lambda\), and \(X\) and \(Y\) are independent, then \(\frac{X}{X + Y}\) has the beta distribution with left parameter \(\alpha\) and right parameter \(\beta\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution inverted beta distribution If \(X\) has the gamma distribution with shape parameter \(\alpha \in (0, \infty)\) and scale parameter \(\lambda \in (0, \infty)\), \(Y\) has the gamma distribution with shape parameter \(\beta \in (0, \infty)\) and scale parameter \(\lambda\), and \(X\) and \(Y\) are independent, then \(\frac{X}{Y}\) has the inverted beta distribution with shape parameters \(\alpha\) and \(\beta\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution Levy distribution If \(X\) has the gamma distribution with shape parameter \(\frac{1}{x}\) and scale parameter \(\sigma \in (0, \infty)\) then \(\frac{1}{X}\) has the Levy distribution with location parameter \(0\) and scale parameter \(\frac{2}{\sigma}\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 geometric distribution geometric distribution If \(X\) has the geometric distributin on \(\{0, 1, \ldots\}\) then \(X + 1\) has the geometric distribution on \(\{1, 2 \ldots\}\). linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 geometric distribution discrete uniform distribution If \(X\) has the geometric distribution on \(\{1, 2, \ldots\}\) with parameter \(p \in (0, 1)\) and \(n \in \{1, 2, \ldots\}\), then the conditional distribution of \(X\) given \(X \in \{1, 2, \ldots, n\}\) converges to the uniform distribution on \(\{1, 2, \ldots, n\}\) as \(p \to 0\). limiting conditional distribution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 geometric distribution exponential distribution If \(X_n\) has the geometric distribution on \(\{1, 2, \ldots\}\) with parmeter \(p_n \in (0, 1)\) for each \(n \in \{1, 2, \ldots\}\) and \(n p_n \to r \in (0, \infty)\) as \(n \to \infty\), then the distribution of \(\frac{X_n}{n}\) converges to the exponential distribution with rate parameter \(r\). limiting distribution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 dinov2008central extreme value distribution extreme value distribution If \(X\) has the Gumbel distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), and \(a \in (-\infty, \infty)\), \(b \in (0, \infty)\), then \(a + b X\) has the Gumbel distribution with location parameter \(a + b \mu\) and scale parameter \(b \sigma\). location-scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Gumbel distribution standard uniform distribution If \(X\) has the Gumbel distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\) then \(\exp\left[-\exp\left(\frac{X - \mu}{\sigma}\right)\right]\) has the standard uniform distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 hypergeometric distribution hypergeometric distribution If \(X\) has the hypergeometric distribution with population size \(m \in \{1, 2, \ldots\}\), sample size \(n \in \{1, 2, \ldots, m\}\) and type parameter \(r \in \{1, 2, \ldots, m\}\) then \(n - X\) has the hypergeometric distribution with population size \(m\), sample size \(n\), and type parameter \(m - r\). linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 hypergeometric distribution binomial distribution Let \(n \in \{1, 2, \ldots\}\) and \(r_m \in \{1, 2, \ldots, m\}\) for each \(m \in \{1, 2, \ldots\}\) with \(\frac{r_m}{m} \to p \in (0, 1)\) as \(m \to \infty\). The hypergeometric distribution with population size \(m\), sample size \(n\), and type parameter \(r_m\) converges to the binomial distribution with trial parameter \(n\) and success parameter \(p\) as \(m \to \infty\). limiting distribution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 hypergeometric distribution Bernoulli distribution If \(X\) has the hypergeometric distribution with population size \(m \in \{1, 2, \ldots\}\), sample size \(n = 1\), and type parameter \(r \in \{1, 2, \ldots, m\}\), then \(X\) has the Bernoulli distribution with parameter \(\frac{r}{m}\). TBD doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 hyperbolic secant distribution continuous uniform distribution If \(X\) has the hyperbolic secant distribution then \(\frac{2}{\pi} \arctan[\exp(\frac{\pi}{2} X)]\) has the standard uniform distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Irwin-Hall distribution Irwin-Hall distribution If \(X\) has the Irwin-Hall distribution with parameter \(m \in \{1, 2, \ldots\}\), \(Y\) has the Irwin-Hall distribution with parameter \(n \in \{1, 2, \ldots\}\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the Irwin-Hall distribution with parameter \(m + n\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Irwin-Hall distribution standard uniform distribution The Irwin-Hall distribution with parameter \(1\) is the standard uniform distribution. special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Irwin-Hall distribution triangular distribution The Irwin-Hall distribution with parmeter \(2\) is the triangular distribution with left endpoint \(0\), right endpoint \(1\) and midpoint \(\frac{1}{2}\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 inverted beta distribution inverted beta distribution If \(X\) has the inverted beta distribution with shape parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) then \(\frac{1}{X}\) has the inverted beta distribution with shape parameters \(\beta\) and \(\alpha\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 inverted beta distribution F-distribution If \(X\) has the inverted beta distribution with shape parameters \(\alpha \in (0, \infty)\) and \(\beta \in (0, \infty)\) then \(\frac{\beta}{\alpha} X\) has the F-distribution with \(2 \alpha\) degrees of freedom in the numerator and \(2 \beta\) degrees of freedom in the denominator. linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Laplace distribution exponential distribution If \(X\) has the Laplace distribution with location parameter \(0\) and scale parameter \(\sigma \in (0, \infty)\) then \(|X|\) has the exponential distribution with scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Levy distribution folded normal distribution If \(X\) has the Levy distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), then \(\frac{1}{\sqrt{X - \mu}}\) has the folded normal distribution with location parameter \(0\) and scale parameter \(\frac{1}{\sigma}\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Levy distribution gamma distribution If \(X\) has the Levy distribution with location parameter \(0\) and scale parameter \(\sigma \in (0, \infty)\), then \(\frac{1}{X}\) has the gamma distribution with shape parameter \(\frac{1}{x}\) and scale parameter \(\frac{2}{\sigma}\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 logarithmic distribution Poisson distribution negative binomial distribution If \((X_1, X_2, \ldots)\) is a sequence of independent random variables, each with the logarithmic distribution with parameter \(p \in (0, 1)\) and \(N\) has the Poisson distribution with parameter \(\lambda \in (0, \infty)\), then \(\sum_{i=1}^N X_i\) has the negative binomial distribution with parameters \(\lambda\) and \(p\). mixture doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 %logistic to standard uniform logistic distribution continuous uniform distribution If \(X\) has the logistic distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\) then \(\frac{1}{1 + \exp\left(\frac{X - \mu}{\sigma}\right)}\) has the standard uniform distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 skew logistic distribution exponential distribution If \(X\) has the skew-logistic distribution with parameter \(\alpha \in (0, \infty)\), then \(Y = \ln(1+e^{-X})\) has the exponential distribution with rate parameter \(\alpha\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 log-normal distribution log-normal distribution If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), \(Y\) has the log-normal distribuiton with location parameter \(\nu \in (-\infty, \infty)\) and scale parameter \(\tau \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(X Y\) has the log-normal distirbution with location parameter \(\mu + \tau\) and scale parameter \(\sqrt{\sigma^2 + \tau^2}\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 log-normal distribution log-normal distribution If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parmaeter \(\sigma \in (0, \infty)\), and \(a \neq 0\) then \(a X\) has the log-normal distribution with location parameter \(a \mu\) and scale parameter \(|a| \sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 log-normal distribution log-normal distribution If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(a \in (0, \infty)\), and \(\sigma \in (0, \infty)\), then \(a X\) has the log-normal distribution with location parameter \(\ln(a) + \mu\) and scale parameter \(\sigma\). linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 log-normal distribution normal distribution If \(X\) has the log-normal distribution with location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\sigma \in (0, \infty)\), then \(\ln(X)\) has the normald distribution with location parameter \(\mu\) and scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Maxwell-Boltzmann distribution Maxwell-Boltzmann distribution If \(X\) has the Maxwell-Boltzmann distribution with scale parameter \(a \in (0, \infty)\) and \(b \in (0, \infty)\), then \(b X\) has the Maxwell-Boltzmann distribution with scale parameter \(a b\). scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Maxwell-Boltzmann distribution chi distribution If \(X\) has the Maxwell-Boltzmann distribution with scale parameter \(a \in (0, \infty)\), then \(\frac{X}{a}\) has the chi distribution with 3 degrees of freedom. scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative binomial distribution negative binomial distribution If \(X\) has the negative binomial distribution with stopping parameter \(r \in (0, \infty)\) and success parameter \(p \in (0, 1)\), \(Y\) has the negative binomial distribution with stopping parameter \(s \in (0, \infty)\) and success parameter \(p\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the negative binomial distribution with stopping parameter \(r + s\) and success parameter \(p\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative binomial distribution geometric distribution The negative binomial distribution with stopping parameter \(1\) and success parameter \(p \in (0, 1)\) is the geometric distribution with success parameter \(p\). Also, if \((X_1, X_2, \ldots, X_n)\) is a sequence of independent Geometric(p) random variables, their sum \(\sum_{i=1}^n{X_i} is negative binomial(n,p). special case convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative binomial distribution Poisson distribution If \(p_r \in (0, 1)\) for each \(r \in (0, \infty)\) and \(r \frac{p}{1-p} \to \lambda \in (0, \infty)\) as \(r \to \infty\), then the negative binomial distribution with stopping parameter \(r\) and success parameter \(p_r\) converges to the Poisson distribution with parameter \(\lambda\). limiting distribution with respect to parameter doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution Exponential distribution If for every t > 0 the number of arrivals in the time interval [0,t] follows the Poisson distribution with mean \(\lambda t\), then the sequence of inter-arrival times are independent and identically distributed exponential random variables with mean \(\fract{1}{λ}\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative binomial distribution normal distribution If \(X\) has the negative binomial distribution with stopping parameter \(r \in (0, \infty)\) and success parameter \(p \in (0, \infty)\), then the distribution of \(\frac{p X - r (1 - p)}{\sqrt{r (1 - p}}\) converges to the standard normal distribution at \(r \to \infty\). central limit theorem doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative binomial distribution binomial distribution For \(k \in \{1, 2, \ldots\}\), let \(Z_k\) denote the number of failures before the \(k\)th success in a sequence of Bernoulli trials with success parameter \(p \in (0, 1)\), so that \(Z_k\) has the negative binomial distribution with stopping parameter \(k\) and success parameter \(p\). Then for \(n \in \{1, 2, \ldots\}\), \(Y_n = \max\{k: k + Z_k \leq n\}\) has the binomial distribution with trial parameter \(n\) and success parameter \(p\). inverse stochastic process doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution log-normal distribution If \(X\) has a normal distribution with mean \(\mu \in (-\infty, \infty)\) and variance \(\sigma^2\), then \(Y = e^X\) has the log-normal distribution with parameters \(\mu\) and \(\sigma^2\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution folded normal Distribution If \(X\) is has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then \(|X|\) has the folded normal distribution with parameters \(\mu\) and \(\sigma\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution half normal distribution If \(X\) is has the normal distribution with mean \(\mu\) = 0 and standard deviation \(\sigma \in (0, \infty)\), then \(|X|\) has a half-normal distribution with parameter \(\sigma\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution non-central chi-square distribution If \(X\) has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then variable \(Y = \frac{X^2}{\sigma^2}\) has a non-central chi-square distribution with one degree of freedom and non-centrality parameter \(\frac{\mu^2}{\sigma^2}\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution truncated normal distribution If \(X\) is has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), and if \(a, b \in [-\infty, \infty]\) with \(a \lt b\), then the conditional distribution of \(X\) given \(X \in (a,b)\) is the truncated normal distribution with location parameter \(\mu\), scale parameter \(\sigma\), minimum value \(a\), and maximum value \(b\). conditioning doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution Levy distribution If \(X\) has the normal distribution with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then \(\frac{1}{(X - \mu)^2}\) has the Levy distribution with location parameter 0 and scale parameter \(\frac{1}{\sigma^2}\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution Rice distribution Let \(\nu \in [0, \infty)\), \(\theta \in (-\infty, \infty)\) and \(\sigma \in (0, \infty)\). If \(X\) has the normal distribution with mean \(\nu \cos(\theta)\) and standard deviation \(\sigma\), \(Y\) has the normal distribution with mean \(\nu \sin(\theta)\) and standard deviation \(\sigma\), and \(X\) and \(Y\) are independent, then \(\sqrt{X^2 + Y^2}\) has the Rice distribution with distance parameter \(\nu\) and scale parameter \(\sigma\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution normal distribution If \(X\) has the normal distribution with mean \(\mu = 0\) and standard deviation \(\sigma = 1\), then \(X\) has a standard normal distribution. special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution chi-square distribution If \(X_1, X_2, \ldots, X_n\) are independent standard normal random variables, then \(\sum_{i=1}^n X_i^2\) has the chi-square distribution with \(n\) degrees of freedom. convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution Students t distribution If \(X_1, X_2, \ldots, X_n\) are independent normally distributed random variables with mean \(\mu \in (-\infty, \infty)\) and standard deviation \(\sigma \in (0, \infty)\), then \(T = \frac{\overline{X} - \mu}{S / \sqrt{n}}\) has the students t distribution with \(n-1\) degrees of freedom. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution Maxwell-Boltzmann distribution If \(X_1\), \(X_2\), and \(X_3\) are independent random variables, each with the normal distribuiton with mean \(0\) and standard deviation \(a \in (0, \infty)\), then \(\sqrt{X_1^2 + X_2^2 + X_3^2}\) has the Maxwell-Boltzmann distribution with parameter \(a\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution Cauchy distribution If \(X\) and \(Y\) are independent variables, each with the standard normal distribution, then \(\frac{X}{Y}\) has the standard Cauchy distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pareto distribution exponential distribution If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), then \(\ln\left(\frac{X}{b}\right)\) has the exponential distribution with rate parameter \(a\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pareto distribution Pareto distribution If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), and \(c \in (0, \infty)\) then \(c X\) has the Pareto distribution with shape parameter \(a\) and scale parameter \(b c\). scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pareto distribution beta distribution If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) then \(\frac{b}{X}\) has the beta distribution with left shape parameter \(a\) and right shape parameter \(1\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pareto distribution continuous uniform distribution If \(X\) has the Pareto distribution with shape parameter \(a \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), then \(1 - \left(\frac{b}{X}\right)^a\) has the standard uniform distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution Poisson distribution If \(X\) has the Poisson distribution with parameter \(\alpha \in (0, \infty)\), \(Y\) has the Poisson distribution with parameter \(\beta \in (0, \infty)\), and \(X\) and \(Y\) are independent, then \(X + Y\) has the Poisson distribution with parameter \(\alpha + \beta\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution normal distribution If \(X\) has the Poisson distribution with parameter \(\alpha \in (0, \infty)\), then the distribution of \(\frac{X - \alpha}{\sqrt{\alpha}}\) converges to the standard normal distribution as \(\alpha \to \infty\). central limit theorem doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution normal distribution As \( \sigma^2=\mu and \mu\to\infty \) Poisson distribution becomes normal distribution. limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution binomial distribution If \(\{N_t: t \ge 0\}\) is a Poisson process and if \(s \lt t\), then the conditional distribution of \(N_s\) given \(N_t = n\) is binomial with parameters \(n\) and \(\frac{s}{t}\). conditioning doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution gamma distribution If \(\{N_t: t \ge 0\}\) is a Poisson process with rate parameter \(\alpha \in (0, \infty)\) and \(n \in \{1, 2, \ldots\}\) then \(T = \min\{t \ge 0: N_t = n\}\) has the gamma distsribution with shape parameter \(k\) and scale parameter \(\frac{1}{\alpha}\). stochastic process doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution logarithmic distribution negative binomial distribution If \(\bs{X} =(X_1, X_2, \ldots)\) is a sequence of independent random variables, each with the logarithmic distribution with parameter \(p \in (0, 1)\), \(N\) has the Poisson distribution with parameter \(-r \ln(1 - p)\) where \(r \in (0, \infty)\), and \(N\) and \(\bs{X}\) are independent, then \(\sum_{i=1}^N X_i\) has the negative binomial distribution with stopping parameter \(r\) and sucess parameter \(p\). compound Poisson transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution gamma distribution negative binomial distribution If \(\Lambda\) has the gamma distribution with shape parameter \(r \in (0, \infty)\) and scale parameter \(\frac{p}{1-p}\) where \(p \in (0, 1)\), and the conditional distribution of \(X\) given \(\Lambda = \lambda \in (0, \infty)\) is Poisson with parameter \(\lambda\), then \(X\) has the negative binomial distribution with stopping parameter \(r\) and success parameter \(p\). mixture doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Rademacher distribution Bernoulli distribution If \(X\) has the Rademacher distribution then \(\frac{X+1}{2}\) has the Bernoulli distribution with success parameter \(\frac{1}{2}\). linear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Rayleigh distribution chi-square distribution If \(X\) has the Rayleigh distribution with scale parameter \(1\), then \(X^2\) has the chi-square distribution with 2 degrees of freedom. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Rayleigh distribution Rayleigh distribution If \(X\) has the Rayleigh distribution with scale parameter \(\sigma \in (0, \infty)\) and \(b \in (0, \infty)\), then \(b X\) has the Rayleigh distribution with scale parameter \(b \sigma\). scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Rayleigh distribution gamma distribution If \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the Rayleigh distribution with scale parameter \(\sigma \in (0, \infty)\), then \(\sum_{i=1}^n X_i^2\) has the chi-square distribution with shape parameter \(n\) and scale parameter \(2 \sigma^2\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Rayleigh distribution continuous uniform distribution If \(X\) has the Rayleigh distribution with shape parameter \(\sigma \in (0, \infty)\), then \(1 - \exp\left(-\frac{X^2}{2 \sigma^2}\right)\) has the standard uniform distribution. nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Rice distribution Rayleigh distribution The Rice distribution with distance parameter \(0\) and scale parameter \(\sigma \in (0, \infty)\) is the Rayleigh distribution with scale parameter \(\sigma\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Rice distribution noncentral chi-square distribution If \(X\) has the Rice distribution with distance parameter \(\nu \in [0, \infty)\) and scale parameter \(1\), then \(X^2\) has the noncentral chi-square distribution with 2 degrees of freedom and noncentrality parameter \(\nu^2\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 semicircle distribution continuous uniform distribution If \(X\) has the semicircle distribution with radius \(r \in (0, \infty)\) then \(\frac{1}{2} + \frac{1}{\pi r^2} X \sqrt{r^2 - X^2} + \frac{1}{\pi} \arcsin\left(\frac{X}{r}\right)\) has the standard uniform distribution nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 stable distribution Cauchy distribution If \(X\) has a stable distribution with stability parameter \(\alpha = 1\), skewness parameter \(\beta = 0\), location parameter \(\mu \in (-\infty, \infty)\), and scale parameter \(\gamma \in (0, \infty)\), then \(X\) has a Cauchy distribution with scale parameter \(\gamma\) and location parameter \(\mu\). special case. doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 stable Distribution normal Distribution If \(X\) has a stable distribution with stability parameter \(\alpha = 2\), location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\gamma \in (0, \infty)\), then \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma^2 = 2 \gamma^2\). special case. doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 stable Distribution Levy Distribution If \(X\) has a stable distribution with stability parameter \(\alpha = \frac{1}{2}\), skewness parameter \(\beta=1\), location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\gamma \in (0, \infty)\), then \(X\) has a Levy distribution with scale parameter \(\gamma\) and shift parameter \(\mu\). special case. doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 stable Distribution Landau Distribution If \(X\) has a stable distribution with stability parameter \(\alpha = 1\), skewness parameter \(\beta = 1\), location parameter \(\mu \in (-\infty, \infty)\) and scale parameter \(\gamma \in (0, \infty)\) then \(X\) has a Landau distribution with scale parameter \(\gamma\) and location parameter \(\mu\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Students t distribution F distribution if \(X\) has the students t-distribution with \(n \in \{1, 2, \ldots\}\) degrees of freedom, then \(Y = X^2\) has the F distribuiton with \(1\) degree of freedom in the numerator and \(n\) degrees of freedom in the denominator. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Students t distribution Cauchy distribution The students t-distribution with 1 degree of freedom is the standard Cauchy distribuiton. special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 U-quadratic distribution continuous uniform distribuiton If \(X\) has the U-quadratic distribution with left endpoint \(a \in (-\infty, \infty)\) and right endpoint \(b \in (a, \infty)\) then \(\frac{\alpha}{3} [(X - \beta)^3 + (\beta - \alpha)^3]\) has the standard uniform distribution, where \(\alpha = \frac{12}{(b - a)^3}\) and \(\beta = \frac{a + b}{2}\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 von Mises distribution continuous uniform distribution The von Mises distribution with location parameter \(0\) and shape parameter \(0\) is the uniform distribution on the interval \([-\pi, \pi]\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Wald distribution Wald distribution If \(X\) has the Wald distribution with mean \(\mu \in (0, \infty)\) and shape parameter \(\lambda \in (0, \infty)\) and \(t \in (0, \infty)\), then \(t X\) has the Wald distribution with mean \(t \mu\) and shape parameter \(t \lambda\) scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Wald distribution Wald distribution If \(X\) has the Wald distribution with mean \(\mu a\) and shape paramter \(\lambda a^2\) where \(\mu \in (0, \infty)\), \(\lambda \in (0, \infty)\), and \(a \in (0, \infty)\), and if \(Y\) has the Wald distribution with mean \(\mu b\) and shape parameter \(\lambda b\) where \(b \in (0, \infty)\), and if \(X\) and \(Y\) are independent, then \(X + Y\) has the Wald distribution with mean \(\mu(a + b)\) and shape paramter \(\lambda(a^2 + b^2)\). convolution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Weibull distribution Weibull distribution If \(X\) has the Weibull distribution with shape parameter \(k \in (0, \infty)\), scale parameter \(b \in (0, \infty)\), and \(c \in (0, \infty)\), then \(Y = c X\) has the Weibull distribution with shape parameter \(k\) and scale parameter \(b c\). scale transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Weibull distribution exponential distribution If \(X\) has the Weibull distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\), then \(Y = \left(\frac{X}{b}\right)^k\) has the standard exponential distribution. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution normal distribution The normal distribution with \(\mu=0\) and \(\sigma^2=1\) is called the standard normal transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 student distribution normal distribution As \(n\longrightarrow\infty\), the t-distribution approaches the normal distribution with mean 0 and variance 1 limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 F distribution student distribution The square root of a Fisher's F distribution is a students t distribution transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 binomial distribution normal distribution If n is large enough, then the skew of the distribution is not too great. In this case, if a suitable continuity correction is used, then an excellent approximation to \(B(n, p)\) is given by the normal distribution \(N(np, np(1-p))\) as \(n \rightarrow \infty\) limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Erlang distribution chi-square distribution When the scale parameter \(\mu\) equals 2, then the Erlang distribution simplifies to the chi-square distribution with \(2k\) degrees of freedom special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 noncentral student t distribution normal distribution If \(T\) is noncentral t-distributed with \(\nu\) degrees of freedom and noncentrality parameter \(\mu\) and \(Z=\lim_{\nu\to\infty}T\), then \(Z\) has a normal distribution with mean \(\mu\) and unit variance limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Pareto distribution If \(X\) has the standard uniform distribution, \(\mu \in (-\infty, \infty)\), and \(\beta \in (0, \infty)\) then \(\frac{\mu}{(1 - X)^{1/\beta}}\) has the Pareto distribution with location parameter \(\mu\) and shape parameter \(\beta\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution exponential distribution If \(X\) has the standard uniform distribution and \(\beta \in (0, \infty)\), then \(-\beta \ln(1 - X)\) has the exponential distribution with scale parameter \(\beta\). nonlinear transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Zipf distribution discrete uniform distribution The discrete uniform distribution is a special case of the Zipf distribution where \(a=0, a=1, b=n\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Fisher-Tippett distribution Gumbel distribution The Gumbel distribution is a particular case of the Fisher-Tippett distribution where \(\mu=0, \beta=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 log-normal distribution Gibrat's distribution Gibrat's law is a special case of the log-normal distribution where special case \(\mu=0, x=1\) doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Cauchy distribution Cauchy distribution If \(X\) is a standard Cauchy distribution, then \(Y = x_0 + \gamma X\) is a Cauchy distribution transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 multinomial distribution Binomial distribution When \(k=2\), the multinomial distribution is the binomial distribution transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 power series distribution Pascal distribution The power series\(c, (A(c))\) distribution becomes a Pascal distribution when \(A(c)=(1-c)^{-x}, c=1-p\) transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 power series distribution logarithmic distribution The power series distribution is a special case of Power series distribution distribution with \(A(c)=-\log (1-c)\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Poisson distribution gamma-Poisson distribution Let \(\mu \sim Gamma(\alpha, \beta)\) denote that \(\mu\) is distributed according to the Gamma density g parameterized in terms of a shape parameter \(\alpha\) and an inverse scale parameter \(\beta\): \(g(\mu \mid \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \mu^{\alpha-1} e^{-\beta \mu}, \mu>0\). Then, given the same sample of n measured values \(k_i\) as before, and a prior of \(Gamma(\alpha, \beta)\), the posterior distribution is \(\mu \sim Gamma(\alpha + \sum_{i=1}^n k_i, \beta + n)\). The posterior predictive distribution of additional data is a Gamma-Poisson distribution. Bayesian doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma-Poisson distribution Pascal distribution The gamma-Poisson distribution becomes a Pascal distribution when \( \alpha=\frac{1-p}{p}, \beta=n \) transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution beta-binomial distribution For \(a = b = 1\), the continuous uniform distribution reduces to the beta-binomial distribution as a special case special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Zipf distribution zeta distribution The zeta distribution is equivalent to the Zipf distribution for infinite N. limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 power series distribution Poisson distribution The power series\((c, A(c))\) distribution becomes a Poisson distribution when \(\mu=c, A(c)=e^c\) transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pascal distribution Poisson distribution Consider a sequence of negative binomial distributions where the stopping parameter n goes to infinity, whereas the probability of success in each trial, p, goes to zero in such a way as to keep the mean of the distribution constant. Denoting this mean \(\mu\), the parameter p will have to be \(\mu = r \frac{p}{1-p} \rightarrow p = \frac{\mu}{r+\mu}\). Then \(Poisson(\mu) = \lim_{n \to \infty} Pascal(n, \frac{\mu}{\mu+n})\). transformation, limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative hypergeometric distribution binomial distribution As \(n_3\to\infty, n_1\to\infty\) and letting \(p=n_1/n_3, n_2=n\), the negative hypergeometric\((n_1, n_2, n_3)\) distribution becomes a binomial\((n, p)\) distribution transformation, limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative hypergeometric distribution Pascal distribution As \(N\to\infty, \frac{K}{N}\to\infty\), the negative hypergeometric\((n_1, n_2, n_3)\) distribution becomes a Pascal\((n, p)\) distribution limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 negative hypergeometric distribution negative binomial distribution As \(N\to\infty, \frac{K}{N}\to\infty\), the negative hypergeometric\((n_1, n_2, n_3)\) distribution becomes a negative binomial\((n, p)\) distribution limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pascal distribution geometric distribution The geometric distribution is a special case of the Pascal distribution where \(n=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 discrete Weibull distribution geometric distribution The geometric distribution is a particular case of the discrete Weibull distribution where \(\beta=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution chi-square distribution If \(X_i \sim Normal(\mu, \sigma^2)\), with \(i=1,...,k\) independent random variables, then \(\sum_{i=1}^{k} (\frac{X_i-\mu}{\sigma})^2\) is a chi-sqaure distribution. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 normal distribution gamma-normal distribution When the \(\sigma\) in the normal distribution is Inverted gamma\((\alpha, \beta)\), the normal distribution becomes a gamma-normal\(\mu, \alpha, \beta)\) distribution Bayesian doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Wald distribution normal distribution As \(\lambda \to \infty\), the Wald distribution becomes more like a standard normal (Gaussian) distribution limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution log-gamma distribution If a random variable \(X\) is gamma-distributed with scale \(\alpha\) and shape \(\beta\), then \(Y = log X\) is log gamma-distributed. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 generalized gamma distribution gamma distribution The gamma distribution is a special case of the generalized gamma distribution where \(\gamma=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Wald distribution chi-square distribution If a random variable \(X\) is inverse Gaussian-distributed with mean \(\mu\) and shape parameter \(\lambda\), the \(Y = \lambda(X-\mu)^2/(\mu^2 X)\) has a chi-square distribution transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution chi-Square distribution If \(X \sim Exponential(\lambda=1/2)\), then \(X \sim \chi_{2}^{2}\) has a chi-square distribution with 2 degrees of freedom transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution Erlang distribution If \(X \sim \chi^{2}(k)\) with even \(k\), then \(X\) is Erlang distributed with shape parameter \(k/2\) and scale parameter \(1/2\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Cauchy distribution arctangent distribution The derivative of the arctangent function gives the formula of the Cauchy distribution. Therefore, the arctangent is called the Cauchy cumulative distribution special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution hypoexponential distribution The hypoexponential distribution is the distribution of a general sum (\(\sum X_i)\) of exponential random variables. Its coefficient of variation is less than one, compared to the exponential distribution, whose coefficient of variation is one transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Makeham distribution Gompertz distribution The Gompertz distribution is a special case of the Makeham distribution where \(\gamma=0\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution F distribution If \(X_1, X_2\) are two independent random variables with exponential distribution with \(\alpha=1\), then \(Y=X_1/X_2\) is an F distribution special case, transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution hyperexponential distribution The hyperexponential distribution is the distribution whose density is a weighted sum of exponential densities. Its coefficient of variation is greater than one, compared to the exponential distribution, whose coefficient of variation is one special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 IDB distribution exponential distribution The exponential distribution is a special case of the IDB distribution where \(\delta=\kappa \to 0\) in the IDB function and \(\alpha=1/ \gamma\) in the exponential function special case, limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Muth distribution exponential distribution The exponential distribution is a particular case of the Muth distribution where \(\alpha=1\) in the exponential function and \(\kappa \to 0\) in the Muth function special case, limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution exponential power distribution If \(X\) has a standard uniform distribution, then \(Y=[log(1-log(1-X))/\gamma]^{1/\kappa}\) has an exponential power distribution with parameters \(\lambda\) and \(\kappa\) transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Laplace distribution error distribution The error distribution is a special case of Laplace distribution where \(\alpha_1=\alpha_2\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution triangular distribution If \(X_1, X_2\) are two independent random variables with standard uniform distribution, then \(X = X_1-X_2\) is a standard triangular distribution transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution power distribution If X is and independent random variable with standard uniform distribution, then \(X^{1/\beta}\) is a standard power distribution transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution power distribution If \(X\) is a standard uniform distribution, then \(X_(n)\) is a standard power distribution special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 IDB distribution Rayleigh distribution The Rayleigh distribution is a special case of the IDB distribution where \(\delta=2/\alpha, \gamma=0\) special case, transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Weibull distribution Rayleigh distribution The Rayleigh distribution is a special case of the Weibull distribution where \(\beta=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 triangular distribution triangular distribution The standard triangular distribution is a special case of the triangular distribution where \(a=-1, b=1, m=0\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 log-logistic distribution Lomax distribution The Lomax distribution is a special case of the Log-logistic distribution where \(\kappa = 1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 log-logistic distribution logistic distribution If X has a log-logistic distribution with scale parameter \(\alpha\) and shape parameter \(\beta\) then \(Y = log(X)\) has a logistic distribution with location parameter \(log(\alpha)\) and scale parameter \(1 / \beta\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Erlang distribution exponential distribution Exponential distribution is a special case of the Erlang distribution where \(n=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution Erlang distribution If X has the exponential distribution with parameter \(\alpha\) then \(\sum^n X_i\) has the Erlang distribution with parameters \(n, \alpha\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 non-central student t distribution student t distribution students t-distribution is a special case of the noncentral students t-distribution where \(\delta=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 logistic exponential distribution exponential distribution exponential distribution is a special case of the logistic exponential distribution distribution where \(\beta=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Benford distribution If X has the standard uniform distribution then \(\lfloor 10^X \rfloor\) has the Benford distribution. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 gamma distribution inverted gamma distribution If X has the gamma distribution then \(\frac{1}{X}\) has the inverted gamma distribution. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Cauchy distribution Cauchy distribution standard Cauchy distribution is a special case of the Cauchy distribution distribution where \(a=0, \alpha=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Cauchy distribution hyperbolic secant distribution If X has the standard Cauchy distribution then \(\log{\frac{\lvert X \rvert}{\pi}}\) has the hyperbolic secant distribution. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 power-series distribuion logarithmic distribution The logarithmic distribution is a particular case of the power-series distribution where \(A(c) = -\log{(1-c)}\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pascal distribuion beta-Pascal distribution When the \(p\propto\beta\) in the pascal distribution, the pascal distribution becomes a beta-pascal Bayesian doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Polya distribuion binomial distribution The binomial distribution is a particular case of the Polya distribution where \(\beta=1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 geometric distribuion Pascal distribution If \(X\) has the geometric distribuion with parameter \(p\), then \(\sum\nolimits_{i=1}^n X_i\) has the pascal distribution with parameters \(n, p\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Pascal distribuion normal distribuion When \(\mu=n(1-p)\) and \(n\to\infty\) then pascal distribuion becomes normal distribuion. limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution normal distribution When \(\beta=\gamma\to\infty\) then beta distribuion becomes normal distribuion. limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 generalized gamma distribution log-normal distribution When \(\beta\to\infty\) then generalized gamma distribuion becomes log-normal distribuion. limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution chi distribution If \(X\) has the chi-square distribuion with parameter \(n\), then \( \sqrt{X} \) has the chi distribution with parameters \(n\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 chi-square distribution exponential distribution The exponential distribution is a special case of chi-square distribution with \( \n=2 \). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution chi-square distribution The chi-square distribution is a special case of exponential distribution with \( \alpha=2 \). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 beta distribution inverted beta distribution If \(X\) has the beta distribuion with parameters \(\beta\) and \(\gamma\), then \( \frac{X}{1-X} \) has the inverted beta distribution with parameters \(\beta\) and \(\gamma\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 hypoexponential distribution Erlang distribution The Erlang distribution is a special case of hypoexponential distribution with \(\bar{\alpha}=\alpha\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 doubly noncentral t distribution noncentral t distribution doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 noncentral f distribution f distribution When \(\delta\to0\) then noncentral f distribuion becomes f distribuion. limiting doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 hyperexponential distribution exponential distribution The exponential distribution is a special case of hyperexponential distribution with \(\bar{\alpha}=\alpha\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 exponential distribution Rayleigh distribution If \(X\) has the exponential distribution with parameter \(\alpha\), then \( X^2 \) has the Rayleigh distribution with parameter \(\alpha\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution Gompertz distribution If \(X\) has the standard uniform distribution, then \( \frac{log{1-\frac{(\log{X})(\log{k})}{\delta}}}{\log{k}} \) has the Gompertz distribution. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 error distribution Laplace distribution The Laplace distribution is a special case of error distribution with \(a=0\), \(b=\frac{\alpha}{2}\), and \(c=2\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 Laplace distribution error distribution The error distribution is a special case of Laplace distribution with \(\alpha_1=\alpha_2\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 continuous uniform distribution uniform distribution If \(X\) has the standard uniform distribution, then \( a+(b-a)X \) has the uniform distribution with parameters \(a\) and \(b\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 standard power distribution continuous uniform distribution The standard uniform distribution is a special case of standard power distribution with \(\beta=1\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 minimax distribution power distribution The standard power distribution is a special case of minimax distribution with \(\gamma=1\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 power distribution power distribution The standard power distribution is a special case of power distribution with \(\alpha=1\). special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 generalized Pareto distribution Pareto distribution If \(X\) has the generalized Pareto distribution with parameters \(\delta\), \(k\), \(\gamma=0\), then \( X+\delta \) has the Pareto distribution with parameters \(k\) and \(\lambda\). transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 extreme-value distribution Weibull distribution If \(X\) has the extreme-value distribution, then \( \log{X} \) has the Weibull distribution with the same parameters. transformation doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 lomax distribution log-logistic distribution The log-logistic distribution is a special case of the lomax distribution where \(\kappa = 1\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448 TSP distribution triangular distribution The triangular distribution is a special case of the TSP distribution where \(n = 2\) special case doi:10.1080/07408170590948512 doi:10.1198/000313008X270448