arcsine distribution Levy arcsine distribution continuous The arcsine distribution models the proportion of time that Brownian motion is positive. $$(0, 1)$$ $$f(x) = \frac{1}{\pi \sqrt{x (1 - x)}}, \; x \in (0, 1)$$ Does not exist $$F(x) = \frac{2}{\pi} \arcsin(\sqrt{x}), \; x \in (0, 1)$$ $$Q(p) = \sin^2(\frac{\pi}{2} p), \; p \in (0, 1)$$ $$\frac{1}{2}$$ $$\frac{1}{8}$$ $$0$$ $$\frac{33}{4}$$ $$\frac{1}{2}$$ $$\frac{2 - \sqrt{2}}{4}$$ $$\frac{2 + \sqrt{2}}{4}$$ Derived by Paul Levy in 1939 as the distribution of proportion of time that Brownian motion is positive. 10.1081/STA-120029820 Bernoulli distribution discrete The Bernoulli distribution governs an indicator random variable. $$p \in [0, 1]$$, the probability of the event $$\{0, 1\}$$ $$f(x) = p^x (1 - p)^{1 - x}, \; x \in \{0, 1\}$$ $$\lfloor 2 p \rfloor$$ $$F(x) = (1 - p)^{1 - x}, \; x \in \{0, 1\}$$ $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$G(t) = 1 - p + p t, \; t \in (-\infty, \infty)$$ $$M(t) = 1 - p + p e^t, \; t \in (-\infty, \infty)$$ $$\varphi(t) = 1 - p + p e^{i t}, \; t \in (-\infty, \infty)$$ $$\mu(n) = p, \; n \in \{0, 1, \ldots\}$$ $$p$$ $$p (1-p)$$ $$\frac{1 - 2 p}{\sqrt{p (1 - p)}}$$ $$\frac{1- 6 p + 6 p^2}{p (1 - p)}$$ $$-(1 - p) \ln(1 - p) - p \ln(p)$$ $$Q(\frac{1}{2})$$ 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 power series exponential Named for Jacob Bernoulli 10.1002/0471722227.fmatter 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 $$(0, 1)$$ $$f(x) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, \; x \in (0, 1)$$ $$\frac{\alpha - 1}{\alpha + \beta - 2}; \; \alpha \in (1, \infty), \beta \in (1, \infty)$$ $$F(x) = \int_0^x f(t) dt$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$, where $$F$$ is the distribution function. $$M(t) = 1 + \sum_{k=1}^\infty \left(\prod_{j=0}^{k-1} \frac{\alpha + j}{\alpha + \beta + j}\right) \frac{t^k}{k!}, \; t \in (-\infty, \infty)$$ $$\frac{\alpha}{\alpha + \beta}$$ $$\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$$ $$\frac{2 (\beta - \alpha) \sqrt{\alpha + \beta + 1}}{(\alpha + \beta + 2) \sqrt{\alpha \beta}}$$ $$\frac{\alpha^3 - \alpha^2 (2 \beta - 1) + \beta^2 (\beta + 1) - 2 \alpha \beta (\beta + 2)}{\alpha \beta (\alpha + \beta + 2)(\alpha + \beta + 3)}$$ $$\ln(B(\alpha, \beta)) - (\alpha - 1) \psi(\alpha) - (\beta - 1) \psi(\beta) + (\alpha + \beta - 2) \psi(\alpha + \beta)$$ where $$\psi$$ is the digamma function $$Q(\frac{1}{2})$$ 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 exponential 10.1080/00949657808810232 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 10.1016/0304-4076(94)01612-4 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 $$\{0, 1, \ldots, n\}$$ $$f(x) = {n \choose x} p^x (1 - p)^{n - x}, \; x \in \{0, 1, \ldots, n\}$$ $$\lfloor (n + 1) p \rfloor$$ $$F(x) = B(1 - p; n - x, x + 1), \; x \in \{0, 1, \ldots, n\}$$ where $$B$$ is the incomplete beta function $$Q(r) = F^{-1}(r), \; r \in [0, 1]$$ where $$F$$ is the distribution function $$G(t) = (1 - p + p t)^n, \; t \in (-\infty, \infty)$$ $$M(t) = (1 - p + p e^t)^n, \; t \in (-\infty, \infty)$$ $$\varphi(t) = (1 - p + p e^{i t})^n, \; t \in (-\infty, \infty)$$ $$n p$$ $$n p (1 - p)$$ $$\frac{1 - 2 p}{\sqrt{n p (1 - p)}}$$ $$\frac{1 - 6 p (1 - p)}{n p (1 - p)}$$ $$\frac{1}{2} \log_2[2 \pi e n p (1 - p)] + O\left(\frac{1}{n}\right)$$ $$Q(\frac{1}{2})$$ where $$Q$$ is the quantile function $$Q(\frac{1}{4})$$ where $$Q$$ is the quantile function $$Q(\frac{1}{3})$$ where $$Q$$ is the quantile function power series exponential The binomial distribution is attributed to Jacob Bernoulli 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 $$\{0, 1, \ldots, n\}$$ $$f(x) = {n \choose x} \frac{B(a + x) B(b + n - x)}{B(a, b)}, \; x \in \{0, 1, \ldots, n\}$$, where $$B$$ is the beta function $$F(x) = \sum_0^x f(t), \quad x \in \{0, 1, \ldots, n\}$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \quad p \in (0, 1)$$ where $$F$$ is the distribution function $$n \frac{a}{a + b}$$ $$n \frac{a b}{(a + b)^2} \frac{a + b + n}{a + b + 1}$$ $$\frac{(a + b + 2n)(b - a)}{(a + b + 2)} \sqrt{\frac{1 + a + b}{b a b (n + a + b)}}$$ $$\frac{(a + b)^2 ( + 1 + b)}{n a b (a + b + 2)(a + b + 3)(a + b + n)} \left[(a + b)(a + b - 1 + 6 n) + 3 a b (n - 2) + 6 n^2 - \frac{3 a b n (6-n)}{a + b} - \frac{18 a b n^2}{(a + b)^2}\right]$$ $$Q(\frac{1}{2})$$ 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 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 $$\alpha \in (-\infty, \infty)$$. the location parameter $$\beta \in (0, \infty)$$, the scale parameter $$(-\infty, \infty)$$ $$f(x) = \frac{1}{\pi} \frac{\beta}{(x - \alpha)^2 + \beta^2}, \; x \in (-\infty, \infty)$$ $$\alpha$$ $$F(x) = \frac{1}{2} + \frac{1}{\pi} \arctan \left(\frac{x - \alpha}{\beta} \right), \; x \in (-\infty, \infty)$$ $$Q(p) = F^{-1}(p) = \alpha + \beta \tan \left(\pi (p - \frac{1}{2}) \right), \; p \in (0, 1)$$ Does not exist $$\varphi(t) = \exp(\alpha i t - \beta |t|), \; t \in (-\infty, \infty)$$ Does not exist Does not exist Does not exist Does not exist $$\ln (4 \pi \beta )$$ $$\alpha$$ $$\alpha - \beta$$ $$\alpha + \beta$$ location scale stable 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. 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 $$(0, \infty)$$ $$f(x) = \frac{1}{2^{n/2} \Gamma(n/2)} x^{n/2-1} e^{-x/2}, \; x \in (0, \infty)$$ $$n-2, \; n \in [2, \infty)$$ $$F(x) = \frac{1}{\Gamma(n/2)} \gamma(x/2; n/2)$$ where $$\gamma$$ is the lower incomplete gamma function $$Q(p) = F^{-1}(p), \; p \in [0, 1)$$ where $$F$$ is the distribution function $$M(t) = \frac{1}{(1 - 2 t)^{n/2}}, \; t \in (-\infty, \frac{1}{2})$$ $$\frac{1}{(1 - 2 i t^{n/2})}, \; t \in (-\infty, \infty)$$ $$n$$ $$2 n$$ $$\sqrt{8/n}$$ $$12/n$$ $$n/2 + \ln(2 \Gamma(n/2)) + (1 - k/2) \psi(n/2)$$ where $$\psi$$ is the digamma 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 quantile function exponential The chi-square distribution was first used by Karl Pearson in 1900. 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 $$(0, \infty)$$ $$f(x) = \frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})$$ $$F(x) = 1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)$$, 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 chi distribution continuous The chi distribution governs the square root of a variable with the chi-square distribtion. $$n \in \{1, 2, \ldots\}$$, the degrees of freedom $$[0, \infty)$$ $$f(x) = \frac{2^{1-n/2}}{\Gamma(n/2)} x^{n-1} e^{-x^2/2}, \; x \in [0, \infty)$$ where $$\Gamma$$ is the gamma function $$\sqrt{n - 1}$$ $$F(x) = \int_0^x f(t) dt, \; x \in$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ 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 $$\sqrt{2} \frac{\Gamma[(n+1)/2]}{\Gamma(n/2)}$$ where $$\Gamma$$ is the gamma function $$n - \mu^2$$ where $$\mu$$ is the mean. $$\frac{\mu}{\sigma^3} (1 - 2 \sigma^2)$$ where $$\mu$$ is the mean and $$\sigma$$ the standard deviation $$\frac{2}{\sigma^2} (1 - \mu \sigma \gamma_1 - \sigma^2)$$ where $$\mu$$ is the mean, $$\sigma$$ the standard deviation, and $$\gamma_1$$ the skewness $$\ln[\Gamma(n/2)] + \frac{1}{2} [n - \ln(2) - (n-1) \psi_0(n/2)]$$ where $$\psi_0$$ is the polygamma function $$Q(\frac{1}{2})$$ 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 continuous uniform distribution rectangular distribution continuous The continuous uniform distribution governs a point chosen at random from an interval. $$a \in (-\infty, \infty)$$, the left endpoint $$b \in (a, \infty)$$, the right endpoint $$[a, b]$$ $$f(x) = \frac{1}{b - a}, \; x \in [a, b]$$ 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 discrete uniform distribution discrete The discrete uniform distribution governs a point chosen at random from an integer interval. $$a \in \{\ldots -2, -1, 0, 1, 2, \ldots\}$$, the left endpoint $$b \in \{a, a+1, \ldots\}$$, the right endpoint $$\{a, a+1, \ldots, b\}$$ $$f(x) = \frac{1}{b - a + 1}, \; x \in \{a, a+1, \ldots, b\}$$ all $$x \in \{a, a+1, \ldots, b\}$$ $$F(x) = \frac{x - a + 1}{b - a + 1}, \; x \in \{a, a+1, \ldots, b\}$$ $$Q(p) = \lceil a + p (b - a) \rceil, \; p \in [0, 1]$$ $$M(t) = \frac{e^{a t} - e^{(b+1) t}}{(b + a + 1)(1 - e^t)}, \; t \in (-\infty, \infty)$$ $$\varphi(t) = \frac{e^{i a t} - e^{i (b+1) t}}{(b + a + 1)(1 - e^{i t})}, \; t \in (-\infty, \infty)$$ $$\frac{1}{2}(a + b)$$ $$\frac{1}{12}[(b - a + 1)^2 - 1]$$ $$0$$ $$-\frac{6}{5} \frac{(b - a + 1)^2 + 1}{(b - a + 1)^2 - 1}$$ $$\ln(a + b - 1)$$ $$\lceil \frac{1}{2}(a + b) \rceil$$ $$\lceil \frac{3}{4} a + \frac{1}{4} b \rceil$$ $$\lceil \frac{1}{4} a + \frac{3}{4} b \rceil$$ exponential distribution negative exponential distribution continuous The exponential distribution models the time between random points in the Poisson model. $$r \in (0, \infty)$$, rate $$[0, \infty)$$ $$f(x) = r e^{-r x}, \; x \in [0, \infty)$$ $$0$$ $$F(x) = 1 - e^{-r x}, \; x \in [0, \infty)$$ $$Q(p) = \frac{- \ln(1 - p)}{r}, \; p \in [0, 1)$$ $$\frac{r}{r - t}, \; t \in (-\infty, t)$$ $$\frac{r}{r - i t}, \; t \in (-\infty, \infty)$$ $$\frac{1}{r}$$ $$\frac{1}{r^2}$$ $$2$$ $$6$$ $$1 - \ln(r)$$ $$\frac{\ln(2)}{r}$$ $$\frac{\ln(4) - \ln(3)}{r}$$ $$\frac{\ln(3)}{r}$$ exponential scale The exponential distribution was named by Karl Pearson in 1895. 10.1017/S0269964807070088 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 $$(0, \infty)$$ $$f(x) = -\frac{1}{p} \frac{b (1 - p) e^{-b x}}{1 - (1 - p) e^{-b x}}, \; x \in [0, \infty)$$ $$0$$ $$F(x) = 1 - \frac{\ln(1 - (1 - p) e^{-b x})}{\ln(p)}, \; x \in [0, \infty)$$ $$Q(r) = \frac{1}{b} \ln\left(\frac{1 - p}{1 - p^{1 - r}}\right), \; r \in (0, 1)$$ $$\mu(n) = -n! \frac{L_{n+1}(1 - p)}{b^n \ln(p)}, \; n \in \{0, 1, \ldots\}$$ where $$L_{n+1}$$ is the polylog function of order $$n + 1$$ $$-\frac{L_2(1 - p)}{b \ln(p)}$$ where $$L_2$$ is the polylog function of order $$2$$. $$-\frac{2 L_3(1 - p)}{b^2 ln(p)} - \frac{L_2^2(1 - p)}{b^2 \ln^2(p)}$$ where $$L_n$$ is the polylog function of order $$n$$ $$\frac{1}{b} \ln(1 +\sqrt{p})$$ $$\frac{1}{b} \ln\left(\frac{1 - p}{1 - p^{3/4}}\right)$$ $$\frac{1}{b} \ln\left(\frac{1 - p}{1 - p^{3/4}}\right)$$ scale 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 $$(-\infty, \infty)$$ $$f(x) = \frac{\beta}{2 \alpha \Gamma(1/\beta)} \exp\left[-\left(\frac{|x - \mu|}{\alpha}\right)^\beta\right], \; x \in (-\infty, \infty)$$ 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 F-distribution Snedecor's F-distribution Fisher-Snedecor distribution continuous The F-distribution governs the ratio of independent, scaled chi-square variables. $$m \in (0, \infty)$$, numerator degrees of freedom $$n \in (0, \infty)$$, denominator degrees of freedom $$(0, \infty)$$ $$f(x) = \frac{1}{x B(m/2, n/2)} \sqrt{\frac{(m x)^m n^n}{(m x + n)^{m+n}}}, \; x \in (0, \infty)$$ where $$B$$ is the beta function $$\frac{m - 2}{m} \frac{n}{n+2}, \; m \in (2, \infty)$$ $$F(x) = \frac{B(m x/(m x + n); m/2, n/2)}{B(m/2, n/2)}$$ where $$B$$ is the beta function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function Does not exist $$\frac{n}{n - 2}, \; n \in (2, \infty)$$ $$\frac{2 n^2 (m + n - 2)}{m (n - 2)^2 (n - 4)}, \; n \in (4, \infty)$$ $$\frac{(2 m + n - 2) \sqrt{8 (n - 4)}}{(n - 6) \sqrt{m (m + n - 2)}}, \; n \in (6, \infty)$$ $$\frac{20 n - 8 n^2 + n^3 + 44 m -32 m n + 5 m n^2 - 22 m^2 - 5 m^2 n - 16}{m (n - 6)(n - 8)(m + n - 2)/12}, \; n \in (8, \infty)$$ $$Q(\frac{1}{2})$$ 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 $$F$$-distribution was first derived by George Snedecor in 1934. The letter F was chosen as a tribute to Ronald Fisher. gamma distribution continuous The gamma distribution governs the arrival times in the Poisson model. $$k \in (0, \infty)$$, the shape parameter $$\theta \in (0, \infty)$$, the scale parameter $$(0, \infty)$$ $$f(x) = \frac{1}{\Gamma(k) \theta^k} x^{k-1} \exp(-\frac{x}{\theta}), \; x \in (0, \infty)$$ where $$\Gamma$$ is the gamma function $$(k - 1) \theta, \; k \in [1, \infty)$$ $$F(x) = \frac{1}{\Gamma(k)} \gamma(k, \frac{x}{\theta}), \; x \in (0, \infty)$$ where $$\Gamma$$ is the gamma function and $$\gamma$$ the lower incomplete gamma function $$Q(p) = F^{-1}(p)$$ where $$F$$ is the distribution function $$M(t) = \frac{1}{(1 - \theta t)^k}, \; t \in (-\infty, 1 / \theta)$$ $$\varphi(t) = \frac{1}{(1 - i \theta t)^k}, \; t \in (-\infty, \infty)$$ $$k \theta$$ $$k \theta^2$$ $$\frac{2}{\sqrt{k}}$$ $$k + \ln(\theta) + \ln(\Gamma(k)) + (1 - k) \psi(k)$$ where $$\psi$$ is the digamma function $$\ln (4 \pi \beta)$$ $$Q(\frac{1}{2})$$ 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 scale exponential 10.1017/S0269964807070088 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 $$\{1, 2, \ldots\}$$ $$f(x) = p (1 - p)^{x - 1}, \; x \in \{1, 2, \ldots\}$$ $$1$$ $$F(x) = 1 - (1 - p)^x, \; x \in \{1, 2, \ldots\}$$ $$Q(r) = \lceil \frac{\ln(1 - r)}{\ln(1 - p)} \rceil, \; r \in [0, 1)$$ $$G(t) = \frac{p t}{1 - (1 - p) t}, \; t \in \left(-\frac{1}{1 - p}, \frac{1}{1 - p}\right)$$ $$M(t) = \frac{p e^t}{1 - (1 - p) e^t}, \; t \in (-\infty, -\ln(1 - p))$$ $$\varphi(t) = \frac{p e^{i t}}{1 - (1 - p) e^{i t}}, \; t \in (-\infty, \infty)$$ $$\frac{1}{p}$$ $$\frac{1 - p}{p^2}$$ $$\frac{2 - p}{\sqrt{1 - p}}$$ $$6 + \frac{p^2}{1 - p}$$ $$-\frac{1}{p}((1 - p) \log_2(1 - p) + p \log_2(p))$$ $$\lceil \frac{-\ln(2)}{\ln(1 - p)} \rceil$$ $$\lceil \frac{\ln(3) - \ln(4)}{\ln(1 - p)} \rceil$$ $$\lceil \frac{-\ln(4)}{\ln(1 - p)} \rceil$$ 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. Gumbel distribution continuous The Gumbel distribution models the limit of of the maximum of independent, identically distributed variables. $$\mu \in (-\infty, \infty)$$, the location parameter $$\sigma \in (0, \infty)$$, the scale parameter $$(-\infty, \infty)$$ $$f(x) = \frac{1}{\sigma} \exp\left(-\frac{x - \mu}{\sigma}\right) \exp\left(-\exp\left(\frac{x - \mu}{\sigma}\right)\right), \; x \in (-\infty, \infty)$$ $$\mu$$ $$F(x) = \exp\left(-\exp\left(\frac{x - \mu}{\sigma}\right)\right), \; x \in (-\infty, \infty)$$ $$Q(p) = \mu - \sigma \ln(-\ln(p)), \; p \in (0, 1)$$ $$M(t) = e^{\mu t} \Gamma(1 - \sigma t), \; t \in (-\infty, \frac{1}{\sigma})$$ $$\mu + \sigma \gamma$$ where $$\gamma$$ is Euler's constant $$\frac{\pi^2}{6} \sigma^2$$ $$\frac{12 \sqrt{6}}{\pi^2} \zeta(3)$$ where $$\zeta$$ is the zeta function $$\frac{12}{5}$$ $$\ln(\sigma) + \gamma + 1$$ where $$\gamma$$ is Euler's constant $$\mu - \sigma \ln(\ln(2))$$ $$\mu - \sigma \ln(\ln(4) - \ln(3))$$ $$\mu - \sigma \ln(\ln(4))$$ location scale The Gumbel distribution is named for Emil Gumbel, who derived it in his study of extreme values in 1954. 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 $$\{\max(0, n + m - N), \ldots, \min(m, n)\}$$ $$f(x) = \frac{{m \choose x} {N-m \choose n-x}}{{N \choose n}}, \; x \in \{\max(0, n + m - N), \ldots, \min(m, n)\}$$ $$\lfloor \frac{(n+1)(m+1)}{N+2} \rfloor$$ $$F(x) = \sum_{\max(0, n+m-N)}^x f(t), \quad x \in \{\max(0, n + m - N), \ldots, \min(m, n)\}$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \quad p \in (0, 1)$$ where $$F$$ is the distribution function $$n \frac{m}{N}$$ $$n \frac{m}{N} \frac{N - m}{N} \frac{N - n}{N - 1}$$ $$\frac{(N - 2 m)(N - 2 n)}{n m (N - m)(N - 2)} \sqrt{\frac{N - 1}{N - 2}}$$ $$\left[ \frac{N^2 (N-1)}{n(N - 2)(N - 3)(N - n)}\right] \left[ \frac{N(N+1) - 6 N(N - n)}{m (N - m)} + \frac{3 n (N - n)(N + 6)}{N^2} - 6 \right]$$ $$Q(\frac{1}{2})$$ 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 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. hyperbolic secant distribution continuous The hyperbolic secant distribution is a symmetric, unimodal distribution but with larger kurtosis than the normal distribution. $$(-\infty, \infty)$$ $$f(x) = \frac{1}{2} \sech\left(\frac{\pi}{2} x\right), \; x \in (-\infty, \infty)$$ $$0$$ $$F(x) = \frac{2}{\pi} \arctan\left[\exp\left(\frac{\pi}{2} x \right)\right], \; x \in (-\infty, \infty)$$ $$Q(p) = \frac{2}{\pi} \ln[\tan(\frac{\pi}{2} p)], \; p \in (0, 1)$$ $$M(t) = \sec(t), \; t \in (-\frac{\pi}{1}, \frac{\pi}{2})$$ $$\varphi(t) = \sech(t), \; t \in (-\infty, \infty)$$ $$0$$ $$1$$ $$0$$ $$2$$ $$\frac{4}{\pi \kappa}$$ where $$\kappa$$ is Catalan's constant $$0$$ $$\frac{2}{\pi} \ln(\sqrt{2} - 1)$$ $$\frac{2}{\pi} \ln(\sqrt{2} + 1)$$ 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 $$[0, n]$$ $$f(x) = \frac{1}{2 (n - 1)!} \sum_{k=0}^n (-1)^k {n \choose k} (x - k)^{n-1} \sgn(x - k), \; x \in [0, n]$$ $$\frac{n}{2}$$ $$F(x) = \int_0^x f(t) dt, \quad x \in [0, n]$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \quad p \in (0, 1)$$ where $$F$$ is the distribution function $$M(t) = \left(\frac{e^t - 1}{t}\right)^n, \; t \in (-\infty, \infty)$$ $$m(t) =$$ $$\frac{n}{2}$$ $$\frac{n^2}{12}$$ $$\frac{n}{2}$$ $$Q(\frac{1}{4})$$ where $$Q$$ is the quantile function $$Q(\frac{3}{4})$$ where $$Q$$ is the quantile function The Irwin-Hall distribution is named for Joseph Irwin and Phillip Hall who independently analyzed the distribution in 1927. 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 $$(0, \infty)$$ $$f(x) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 + x)^{-(\alpha + \beta)}, \; x \in (0, \infty)$$ $$\frac{\alpha - 1}{\beta + 1}$$ 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 $$\beta \in (2, \infty)$$ $$Q(\frac{1}{2})$$ 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 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 $$\mu \in (-\infty, \infty)$$, location $$b \in (0, \infty)$$, scale $$(-\infty, \infty)$$ $$f(x) = \frac{1}{2 b} \exp\left(-\frac{|x - \mu|}{b}\right), \; x \in (-\infty, \infty)$$ $$\mu$$ $$F(x) = \frac{1}{2} \exp\left(\frac{x - \mu}{b}\right), \; x \in (-\infty, \mu]; \quad F(x) = 1 - \frac{1}{2} \exp\left(-\frac{x - \mu}{b}\right), \; x \in [\mu, \infty)$$ $$Q(p) = \mu + b \ln(2 \min\{p, 1 - p\}), \; p \in (0, 1)$$ $$M(t) = \frac{e^{\mu t}}{1 - b^2 t}, \; t \in (-\frac{1}{b}, \frac{1}{b})$$ $$\varphi(t) = \frac{e^{\mu i t}}{1 + b^2 t}, \; t \in (-\infty, \infty)$$ $$\mu$$ $$2 b^2$$ $$0$$ $$3$$ $$\log(2 e b)$$ $$\mu$$ $$\mu - b \ln(2)$$ $$\mu + b \ln(2)$$ location scale The Laplace distribution is named for Pierre Simon Laplace. Levy distribution van der Waals profile stable distribution continuous The Levy distribution is a stable distribution that has applications in spectroscopy. $$\mu \in (-\infty, \infty)$$, the location parameter $$c \in (0, \infty)$$, the scale parameter $$(\mu, \infty)$$ $$f(x) = \sqrt{\frac{c}{2 \pi}} \frac{e^{-c/2(x - \mu)}}{(x - \mu)^{3/2}}, \; x \in (\mu, \infty)$$ $$\mu + \frac{c}{3}$$ $$F(x) = \int_\mu^x f(t) dt, \; x \in$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$\varphi(t) = \exp(i \mu t - \sqrt{-2 i c t}), \; t \in (-\infty, \infty)$$ $$\infty$$ undefined undefined undefined $$\frac{1}{2}[1 + 3 \gamma + \ln(16 \pi c^2)]$$ where $$\gamma$$ is Euler's constant $$Q(\frac{1}{2})$$ 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 location scale stable The Levy distribution is named for Paul Pierre Levy. 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 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 $$\{1, 2, \ldots\}$$ $$f(x) = \frac{-1}{\ln(1 - p)} \frac{p^x}{x}, \; x \in \{1, 2, \ldots\}$$ $$1$$ $$F(x) = \sum_1^x f(t), \quad x \in \{1, 2, \ldots\}$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \quad p \in (0, 1)$$ where $$F$$ is the distribution function $$G(t) = \frac{\ln(1 - p t)}{\ln(1 - p)}, \; t \in (-\frac{1}{p}, \frac{1}{p})$$ $$M(t) = \frac{\ln(1 - p e^t)}{\ln(1 - p)}, \; t \in (-\infty, -\ln(p))$$ $$\varphi(t) = \frac{\ln(1 - p e^{i t})}{\ln(1 - p)}, \; t \in (-\infty, \infty)$$ $$\frac{-1}{\ln(1 - p)} \frac{p}{1 - p}$$ $$-\frac{\ln(1 - p)}{\ln^2(1 - p)} \left(\frac{p}{1 - p}\right)^2$$ $$Q(\frac{1}{2})$$ 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 power series The logarithmic distribution was first derived by Ronald Fisher in 1943. logistic distribution continuous The logistic distribution occurs in logistic regression. $$\mu \in (-\infty, \infty)$$, the location parameter $$b \in (0, \infty)$$, the scale parameter $$(-\infty, \infty)$$ $$f(x) = \frac{\exp\left[-\left(\frac{x - \mu}{b}\right)\right]}{b\left\{1 + \exp\left[-\left(\frac{x - \mu}{b}\right)\right]\right\}^2}, \; x \in (-\infty, \infty)$$ $$\mu$$ $$F(x) = \frac{1}{1 + \exp\left[-\left(\frac{x - \mu}{b}\right)\right]}, \; x \in (-\infty, \infty)$$ $$Q(p) = \mu + b \ln\left(\frac{p}{1 - p}\right), \; p \in (0, 1)$$ $$M(t) = e^{\mu t} B(1 - b t, 1 + b t)$$ where $$B$$ is the beta function $$\mu$$ $$\frac{\pi^2}{3} b^2$$ $$0$$ $$\frac{6}{5}$$ $$\ln(b) + 2$$ $$\mu$$ $$\mu - \ln(3) b$$ $$\mu + \ln(3) b$$ location scale Logistic regression was first used by D.R. Cox in 1958. 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 $$(-\infty, \infty)$$ $$f(x;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}\frac{\exp(-\beta x)} {(1+\exp(-x))^{\alpha+\beta}}$$ log-normal distribution log normal distribution lognormal distribution Galton distribution continuous The log-normal distribution models certain skewed variables. $$\mu \in (-\infty, \infty)$$, the location parameter $$\sigma \in (0, \infty)$$, the scale parameter $$(0, \infty)$$ $$f(x) = \frac{1}{x \sqrt{2 \pi \sigma^2}} \exp\left[-\frac{1}{2}\left(\frac{ln(x) - \mu}{\sigma}\right)^2\right], \; x \in (0, \infty)$$ $$e^{\mu - \sigma^2}$$ $$F(x) = \Phi \left(\frac{ln(x) - \mu}{\sigma}\right), \; x \in (0, \infty)$$ where $$\Phi$$ is the standard normal distribution function $$Q(p) = \exp\left(\mu + \sigma \Phi^{-1}(p)\right)$$, where $$\Phi$$ is the standard normal distribution function $$\mu(n) = \exp(\mu n + \frac{1}{2} \sigma^2 n^2), \; n \in \{0, 1, \ldots\}$$ $$\exp(\mu + \frac{1}{2} \sigma^2)$$ $$\exp\left(2 (\mu + \sigma^2)\right) - \exp(2 \mu + \sigma^2)$$ $$\left(e^{\sigma^2} - 2\right) \sqrt{e^{\sigma^2} - 1}$$ $$e^{4 \sigma^2} + 2 e^{3 \sigma^2} + 3 e^{2 \sigma^2} - 6$$ $$\frac{1}{2}[1 + \ln(2 \pi \sigma^2)] + \mu$$ $$e^\mu$$ $$\exp\left(\mu + \sigma \Phi^{-1}(\frac{1}{4})\right)$$, where $$\Phi$$ is the standard normal distribution function $$\exp\left(\mu + \sigma \Phi^{-1}(\frac{3}{4})\right)$$, 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. log-logistic distribution Fisk distribution continuous The log-logistic distribution models lifetimes of devices whose failure rates at first increase and then decrease. $$\alpha \in (0, \infty)$$, the scale parameter $$\beta \in (0, \infty)$$, the shape parameter $$(0, \infty)$$ $$f(x) = \frac{\beta}{\alpha} \frac{(x/\alpha)^{\beta-1}}{[1 + (x/\alpha)^\beta]^2}, \; x \in (0, \infty)$$ $$0$$, if $$\beta = 1$$; $$\alpha \left(\frac{\beta - 1}{\beta + 1}\right)^{1/\beta}$$, if $$\beta \in (1, \infty)$$ $$F(x) = \frac{1}{1 + (x/\alpha)^{-\beta}}, \; x \in (0, \infty)$$ $$Q(p) = \alpha \left(\frac{p}{1 - p}\right)^{1/\beta}, \; p \in (0, 1)$$ $$\mu(n) = \alpha^n \frac{n \pi /\beta}{\sin(n \pi / \beta)}, \; n \lt \beta$$ $$\frac{\alpha \pi / \beta}{\sin(\pi / \beta)}$$ if $$\beta \in (1, \infty)$$ $$\alpha^2 \left[\frac{2 \pi / \beta}{\sin(2 \pi / \beta)} - \frac{(\pi / \beta)^2}{\sin^2(\pi / \beta)}\right]$$ if $$\beta \in (2, \infty)$$ $$\alpha$$ $$\alpha \left(\frac{1}{3}\right)^{1/\beta}$$ $$\alpha 3^{1/\beta}$$ scale The log-logistic distribution is known as the Fisk distribution by economists. P.R. Fisk used the distribution to model income in 1961. Maxwell-Boltzmann distribution continuous The Maxwell-Boltzmann Distribution arises in the kinetic theory of gases. $$a \in (0, \infty)$$, the scale parameter $$[0, \infty)$$ $$f(x) = \sqrt{\frac{2}{\pi}} \frac{1}{a^3} x^2 e^{-x^2/2 a}, \; x \in [0, \infty)$$ $$\sqrt{2} a$$ $$F(x) = \int_a^x f(t) dt, \; x \in$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$2 a \sqrt{\frac{2}{\pi}}$$ $$a^2 \frac{3 \pi - 8}{\pi}$$ $$\frac{2 \sqrt{2}(16 - 5 \pi)}{(3 \pi - 8)^{3/2}}$$ $$4 \frac{-96 + 40 \pi - 3\pi^2}{(3 \pi - 8)^2}$$ $$Q(\frac{1}{2})$$ 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 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. 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 $$\{k, k+1, \ldots\}$$ $$f(x) = {x-1 \choose k-1} p^x (1 - p)^{x-k}, \; x \in \{k, k+1, \ldots\}$$ $$\lfloor 1 + \frac{k-1}{p}\rfloor$$ $$F(x) = \sum_{j=k}^x f(j) , \; x \in \{k, k+1, \ldots\}$$ 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. 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 $$(-\infty, \infty)$$ $$f(x)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{1}{2}(\frac{x - \mu}{\sigma})^2 \right), \; x \in (-\infty, \infty)$$ $$\mu$$ $$F(x) = \Phi\left(\frac{x - \mu}{\sigma}\right), \; x \in (-\infty, \infty)$$ 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 $$M(t) = \exp(\mu t + \frac{1}{2} \sigma^2 t^2), \; t \in (-\infty, \infty)$$ $$\varphi(t) = \exp(i \mu t - \frac{1}{2} \sigma^2 t^2), \; t \in (-\infty, \infty)$$ $$\mu$$ $$\sigma^2$$ $$0$$ $$0$$ $$\frac{1}{2} \ln(2 \pi e \sigma^2)$$ $$\mu$$ $$\mu - \Phi^{-1}(\frac{1}{4}) \sigma$$ where $$\Phi$$ is the standard normal distribution function $$\mu + \Phi^{-1}(\frac{1}{4}) \sigma$$ where $$\Phi$$ is the standard normal distribution function location scale exponential stable 10.1016/j.compedu.2006.06.003 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. Student's t distribution Student t distribution Students t distribution continuous Student's t distirbutiuon 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 Student’s 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 Student’s t-distribution is a special case of the generalised hyperbolic distribution $$df \in (1, \infty)$$, degrees of freedom $$(-\infty, \infty)$$ $$\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\, \Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\! \; x \in (-\infty, \infty)$$ $$\mu$$ $$\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right) \cdot \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2}; -\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)} \end{matrix}$$, where $${ }_2F_1$$ is the hypergeometric function $$M(t) = \exp(\mu t + \frac{1}{2} \sigma^2 t^2), \; t \in (-\infty, \infty)$$ 0 $$\frac{\nu}{\nu-2},$$ for $$\nu > 2; \infty$$ for $$1 \lt \nu \leq 2$$, otherwise undefined 0 $$\frac{6}{\nu-4},$$ for $$\nu > 4$$ 0 location scale exponential stable 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 $$[a, b]$$ $$f(x;\mu,\sigma, a,b) = \frac{1}{\sigma Z}\phi(\xi)$$, where $$\xi=\frac{x-\mu}{\sigma},\ \alpha=\frac{a-\mu}{\sigma},\ \beta=\frac{b-\mu}{\sigma}$$, and $$Z=\Phi(\beta)-\Phi(\alpha)$$ $$\left\{\begin{array}{ll}a, if \mu \lt a \\ \mu, if a\le\mu\le b\\ b, if \mu \gt b \end{array}\right.$$ $$F(x;\mu,\sigma, a,b) = \frac{\Phi(\xi) - \Phi(\alpha)}{Z}$$ $$\mu + \frac{\phi(\alpha)-\phi(\beta)}{Z}\sigma$$ $$\sigma^2\left[1+\frac{\alpha\phi(\alpha)-\beta\phi(\beta)}{Z} -\left(\frac{\phi(\alpha)-\phi(\beta)}{Z}\right)^2\right]$$ Pareto distribution Bradford distribution continuous The Pareto distribution models highly skewed variables that sometimes arise in economics. $$k \in (0, \infty)$$, the shape parameter $$a \in (0, \infty)$$, the scale parameter $$[a, \infty)$$ $$f(x) = \frac{k a^k}{x^{k+1}}, \; x \in [a, \infty)$$ $$a$$ $$F(x) = 1 - \left(\frac{a}{x}\right)^a, \; x \in [a, \infty)$$ $$Q(p) = \frac{a}{(1 - p)^{1/k}}, \; p \in [0, 1)$$ where $$F$$ Does not exist $$\varphi(t) = k (-i a t)^k \gamma(-k, -i a t)$$ where $$\gamma$$ is the lower incomplete gamma function $$\mu(t) = a^t \frac{k}{k - t}, \; t \in (0, k)$$ $$a \frac{k}{k - 1}$$ $$a^2 \frac{k}{(k - 1)^2 (k - 2)}$$ $$\frac{2 (1 + k)}{k - 3} \sqrt{\frac{k - 2}{k}}, \; k \in (3, \infty)$$ $$\frac{6 (k^3 + k^2 - 6 k - 2}{k (k - 3) (k - 4)}$$ $$\ln\left(\frac{k}{a}\right) - \frac{1}{k} - 1$$ $$a 2^{1/k}$$ $$a (\frac{4}{3})^{1/k}$$ $$a 4 ^{1/k}$$ scale The Pareto distributin is named for the Italian economist Vilfredo Pareto, who used the distribution to model wealth, income and other economic variables. 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 $$\{0, 1, 2, \ldots\}$$ $$f(x) = e^{-\lambda} \frac{\lambda^k}{k!}, \; k \in \{0, 1, 2, \ldots\}$$ $$\lfloor \lambda \rfloor$$ $$F(x) = \frac{\gamma(x + 1, \lambda)}{x!}, \; x \in \{0, 1, 2, \ldots\}$$ where $$\gamma$$ is the lower incomplete gamma function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$G(t) = e^{\lambda (t - 1)}, \; t \in (-\infty, \infty)$$ $$M(t) = \exp(\lambda(e^t - 1)), \; t \in (-\infty, \infty)$$ $$\varphi(t) = \exp(\lambda(e^{i t} - 1)), \; t \in (-\infty, \infty)$$ $$m(k) = \lambda^k, \; k \in \{0, 1, 2, \ldots\}$$ $$\lambda$$ $$\lambda$$ $$\sqrt{\lambda}$$ $$\frac{1}{\lambda}$$ $$\lambda [1 - \log(\lambda)] + e^{-\lambda} \sum_{k=0}^\infty \frac{\lambda^k \log(k!)}{k!}$$ $$Q(\frac{1}{2})$$ 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 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. Rademacher distribution discrete The Rademacher distribution arises in physics and in bootstrapping. $$\{-1, 1\}$$ $$f(x) = \frac{1}{2}, \; x \in \{-1, 1\}$$ $$\{-1, 1\}$$ $$F(-1) = \frac{1}{2}, \; F(1) = 1$$ $$Q(p) = -1, \; p \in [0, \frac{1}{2}]; \quad Q(p) = 1, \; p \in (\frac{1}{2}, 1]$$ $$M(t) = \cosh(t), \; t \in (-\infty, \infty)$$ $$M(t) = \cos(t), \; t \in (-\infty, \infty)$$ $$\mu(n) = 1, \; n \in \{0, 2, \ldots\}; \quad \mu(n) = 0, \; n \in \{1, 3, \ldots\}$$ $$0$$ $$1$$ $$0$$ $$-2$$ $$\ln(2)$$ $$0$$ $$-1$$ $$1$$ The Rademacher distribution is named for the German mathematician Hans Rademacher. 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 $$[0, \infty)$$ $$f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2 \sigma^2}\right), \; x \in [0, \infty)$$ $$\sigma$$ $$F(x) = 1 - \exp\left(-\frac{x^2}{2 \sigma^2}\right)$$ $$Q(p) = \sigma \sqrt{-2 \ln(1 - p)}, \; p \in [0, 1)$$ $$M(t) = 1 + \sqrt{\frac{\pi}{2}}\sigma t \exp\left(\frac{1}{2} \sigma^2 t^2\right)\left[\erf\left(\frac{1}{\sqrt{2}} t \right) + 1\right], \; t \in (-\infty, \infty)$$ where $$\erf$$ is the error function $$\mu(t) = \Gamma\left(1 + \frac{t}{2}\right), \; t \in [0, \infty)$$ $$\sigma \sqrt{\frac{\pi}{2}}$$ $$\frac{4 - \pi}{2} \sigma^2$$ $$\frac{2 \sqrt{\pi} (\pi - 3)}{(4 - \pi)^{3/2}}$$ $$-\frac{6 \pi^2 - 24 \pi + 16}{(4 - \pi)^{3/2}}$$ $$1 + \ln\left(\frac{\sigma}{\sqrt{2}}\right) + \frac{\gamma}{2}$$ 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). 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 $$[0, \infty)$$ $$f(x) = \frac{x}{\sigma^2} \exp\left[-\frac{(x^2 + \nu^2)}{2 \sigma^2} \right] I_0\left(\frac{x \nu}{\sigma^2}\right), \; x \in [0, \infty)$$ where $$I_0$$ is the modified Bessel function. $$F(x) = 1 - Q\left(\frac{\nu}{\sigma}, \frac{x}{\sigma}\right), \; x \in [0, \infty)$$ where $$Q$$ is the Marcum $$Q$$-function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$\sigma \sqrt{\frac{\pi}{2}} L_{1/2}\left(-\frac{\nu^2}{2 \sigma^2}\right)$$ where $$L_{1/2}$$ is the Laguerre polynomial of order $$1/2$$. $$2 \sigma^2 + \nu^2 - \frac{\pi \sigma^2}{2} L^2_{1/2}\left(-\frac{\nu^2}{2 \sigma^2}\right)$$ where $$L_{1/2}$$ is the Laguerre polynomial of order $$1/2$$. $$Q(\frac{1}{2})$$ 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 scale The Rice distribution is named for Stephen O. Rice who used the distribution in 1945 in his study of random noise. semicircle distribution Wigner distribution Stato-Tate distribution continuous The semicircle distribution arises as the limiting distribution of the eigenvalues of random symmetric matrices. $$r \in (0, \infty)$$, the radius $$[-r, r]$$ $$f(x) = \frac{2}{\pi r^2} \sqrt{r^2 - x^2}, \; x \in [-r, r]$$ $$0$$ $$F(x) = \frac{1}{2} + \frac{1}{\pi r^2} x \sqrt{r^2 - x^2} + \frac{1}{\pi} \arcsin\left(\frac{x}{r}\right), \; x \in [-r, r]$$ $$Q(p) = F^{-1}(p), \quad p \in (0, 1)$$ where $$F$$ is the distribution function $$M(t) = 2 \frac{I_1(r t)}{r t}, \; t \in (-\infty, \infty)$$ where $$I_1$$ is the modified Bessel function $$\varphi(t) = 2 \frac{J_1(r t)}{r t}, \; t \in (-\infty, \infty)$$ where $$J_1$$ is the Bessel function $$\mu(n) = 0, \; n \in \{1, 3, \ldots\}; \quad \mu(n) = \frac{1}{n + 1} {2 n \choose n}, \; n \in \{0, 2, \ldots\}$$ $$0$$ $$\frac{r^2}{4}$$ $$0$$ $$-1$$ $$\ln(\pi r) - \frac{1}{2}$$ $$0$$ $$Q(\frac{1}{4})$$ where $$Q$$ is the quantile function $$Q(\frac{3}{4})$$ where $$Q$$ is the quantile function 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. standard Cauchy distribution continuous The standard Cauchy distribution governs the ratio of two independent, standard normal variables. $$(-\infty, \infty)$$ $$f(x) = \frac{1}{\pi (1 + x^2)}, \; x \in (-\infty, \infty)$$ $$0$$ $$F(t) = \frac{1}{2} + \frac{1}{\pi} \arctan(x), \; x \in (-\infty, \infty)$$ $$Q(p) = \tan(\pi (p - \frac{1}{2})), \; p \in (0, 1)$$ Does not exist $$\varphi(t) = e^{-|t|}$$ Does not exist Does not exist Does not exist Does not exist $$\ln( 4 \pi)$$ $$0$$ $$-1$$ $$1$$ standard Gumbel distribution continuous The standard Gumbel distribution models the limit of of the maximum of independent, identically distributed variables. $$(-\infty, \infty)$$ $$f(x) = e^{-x} e^{-e^{-x}}, \; x \in (-\infty, \infty)$$ $$0$$ $$F(x) = e^{-e^{-x}}, \; x \in (-\infty, \infty)$$ $$Q(p) = -\ln(-\ln(p)), \; p \in (0, 1)$$ $$M(t) = \Gamma(1 - t), \; t \in (-\infty, 1)$$ $$\gamma$$ where $$\gamma$$ is Euler's constant $$\frac{\pi^2}{6}$$ $$\frac{12 \sqrt{6}}{\pi^2} \zeta(3)$$ where $$\zeta$$ is the zeta function $$\frac{12}{5}$$ $$\gamma + 1$$ where $$\gamma$$ is Euler's constant $$-\ln(\ln(2))$$ $$-\ln(\ln(4) - \ln(3))$$ $$-\ln(\ln(4))$$ standard logistic distribution continuous The standard logistic distribution arises in logistic regression $$(-\infty, \infty)$$ $$f(x) = \frac{e^x}{(1 + e^x)^2}, \; x \in (-\infty, \infty)$$ $$0$$ $$F(x) = \frac{e^x}{1 + e^x}, \; x \in (-\infty, \infty)$$ $$Q(p) = \ln\left(\frac{p}{1 - p}\right), \; p \in (0, 1)$$ $$M(t) = B(1 - t, 1 + t), \; t \in (-\infty, \infty)$$ where $$B$$ is the beta function $$0$$ $$\frac{\pi^2}{3}$$ $$0$$ $$\frac{6}{5}$$ $$\ln(2)$$ $$0$$ $$-\ln(3)$$ $$\ln(3)$$ standard normal distribution continuous The standard normal distribution models standardized physical quantities subject to numerous small, random errors. $$(-\infty, \infty)$$ $$f(x) = \frac{1}{\sqrt{2 \pi}} \exp\left(-\frac{1}{2} x^2\right), \; x \in (-\infty, \infty)$$ $$0$$ $$F(x) = \int_{-\infty}^x f(t) dt, \; x \in (-\infty, \infty)$$ where $$f$$ is the density function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$M(t) = \exp(\frac{1}{2} t^2), \; t \in (-\infty, \infty)$$ $$\varphi(t) = \exp(-\frac{1}{2} t^2), \; t \in (-\infty, \infty)$$ $$0$$ $$1$$ $$0$$ $$0$$ $$\frac{1}{2} \ln(2 \pi)$$ $$0$$ $$Q(\frac{1}{4})$$ where $$Q$$ is the quantile function $$Q(\frac{3}{4})$$ where $$Q$$ is the quantile function The term standard normal distribution came into general use around 1950. standard uniform distribution continuous The standard uniform distribution models a point chosen at random from the interval $$[0, 1]$$. $$[0, 1]$$ $$f(x) = 1, \; x \in [0, 1]$$ $$F(x) = x, \; x \in [0, 1]$$ $$F(x) = \int_a^x f(t) dt, \quad x \in$$ where $$f$$ is the probability density function $$Q(p) = p, \quad p \in [0, 1]$$ $$M(t) = \frac{e^t - 1}{t}, \; t \in (-\infty, \infty)$$ $$\varphi(t) = \frac{e^{i t} - 1}{it}, \; t \in (-\infty, \infty)$$ $$\mu(n) = \frac{1}{n+1}, \; n \in {0, 1, \ldots}$$ $$\frac{1}{2}$$ $$\frac{1}{12}$$ $$0$$ $$-\frac{6}{5}$$ $$0$$ $$\frac{1}{2}$$ $$\frac{1}{4}$$ $$\frac{3}{4}$$ t-distribution Student t-distribution continuous The Student t-distribution arises in a sample from a normal distribution, when the sample mean is standardized using the sample standard deviation $$n \in (0, \infty)$$, degrees of freedom $$(-\infty, \infty)$$ $$f(x) = \frac{\Gamma((n+1)/2)}{\sqrt{n \pi} \Gamma(n/2)} \left( 1 + \frac{x^2}{n} \right)^{-(n+1)/2}, \; x \in (-\infty, \infty)$$ where $$\Gamma$$ is the gamma function $$0$$ $$F(x) = \frac{B(x; n/2, n/2)}{B(n/2, n/2)}$$ where $$B$$ is the beta function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function Does not exist $$\frac{K_{n/2}(\sqrt{n} |t|) (\sqrt{n} |t|)^{n/2}}{\Gamma(n/2) 2^{n/2-1}}, \; t -in (-\infty, \infty)$$ where $$K_n$$ is the Bessel function and $$\Gamma$$ is the gamma function $$0, \; n \in (1, \infty)$$ $$\frac{n}{n-2}, \; n \in (2, \infty)$$ $$0, \; n \in (3, \infty)$$ $$\frac{12}{n}$$ $$\frac{n}{2} + \ln(2 \Gamma(\frac{n}{2})) + (1 - \frac{k}{2}) \psi(\frac{n}{2})$$ where $$\psi$$ is the digamma 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 quantile function The $$t$$-distribution was derived by William Sealy Gosset while he worked at the Guinness Brewery in Dublin. Gosset published his work under the pseudonym Student. The $$t$$-test in statistics was developed by Ronald Fisher who called the distribution Student's distribution triangular distribution continuous The triangular distribution arises from various simple combinations of continuous uniform distributions. $$a \in (-\infty, \infty)$$, the left endpoint $$b \in (a, \infty)$$, the right endpoint $$c \in [a, b]$$, the mode $$[a, b]$$ $$f(x) = \frac{2 (x - a)}{(b - a)(c - a)}, \; x \in [a, c]; \quad f(x) = \frac{2 (b - x)}{(b - a)(b - c)}, \; x \in [c, b]$$ $$c$$ $$F(x) = \frac{(x - a)^2}{(b - a)(c - a)}, \; x \in [a, c]; \quad F(x) = 1 - \frac{(b - x)^2}{(b - a)(b - c}, \; x \in [c, b]$$ $$Q(p) = a + \sqrt{(b - a)(c - a) p}, \; p \in \left[0, \frac{c - a}{b - a}\right]; \quad Q(p) = b - \sqrt{(1 - p)(b - a)(b - c)}, \; p \in \left[\frac{c - a}{b - a}, 1\right]$$ $$M(t) = 2 \frac{(b - c) e^{a t} - (b - a) e^{c t} + (c - a) e^{b t}}{(b - a)(c - a)(b - c) t^2}, \; t \in (-\infty, \infty)$$ $$\frac{a + b + c}{3}$$ $$\frac{a^2 + b^2 + c^2 - a b - a c - b c}{18}$$ $$\frac{\sqrt{2} (a + b - 2 c)(2 a - b - c)(a - 2 b + c)}{5(a^2 + b^2 + c^2 - a b - a c - b c)}$$ $$-\frac{3}{5}$$ $$\frac{1}{2} + \ln\left(\frac{b - a}{2}\right)$$ $$a + \sqrt{\frac{1}{2}(b - a)(c - a)}$$ if $$c \geq \frac{a + b}{2}$$; $$b - \sqrt{\frac{1}{2}(b - a)(b - c)}$$ if $$c \leq \frac{a + b}{2}$$ $$a + \sqrt{\frac{1}{4}(b - a)(c - a)}$$ if $$c \geq \frac{3}{4} a + \frac{1}{4} b$$; $$b - \sqrt{\frac{3}{4}(b - a)(b - c)}$$ if $$c \leq \frac{3}{4} a + \frac{1}{4} b$$ $$a + \sqrt{\frac{3}{4}(b - a)(c - a)}$$ if $$c \geq \frac{1}{4} a + \frac{3}{4} b$$; $$b - \sqrt{\frac{1}{4}(b - a)(b - c)}$$ if $$c \leq \frac{1}{4} a + \frac{3}{4} b$$ U-quadratic distribution continuous the U-quadratic distribution models certain symmetric, bimodal variables. $$a \in (-\infty, \infty)$$, the left endpoint $$b \in (a, \infty)$$, the right endpoint $$[a, b]$$ $$f(x) = \frac{12}{(b - a)^3} \left(x - \frac{a + b}{2}\right)^2, \; x \in [a, b]$$ $$\{a, b\}$$ $$F(x) = \frac{4}{(b - a)^3} \left[\left(x - \frac{a+b}{2}\right)^3 + \left(\frac{a+b}{2} - \frac{12}{(b - a)^3} \right)^3 \right], \; x \in [a, b]$$ $$Q(p) = F^{-1}(p), \quad p \in (0, 1)$$ where $$F$$ is the distribution function $$M(t) = \frac{3 e^{a t}[4 + (a^2 + 2 a (b - 2) + b^2) t] - 3 e^{b t}[4 + ((a+b)^2 - 4 b) t]}{(b - a)^3 t^2}, \; t \in (-\infty, \infty)$$ $$\frac{a+b}{2}$$ $$\frac{3}{20}(b - a)^3$$ $$0$$ $$\frac{3}{112}(b - a)^4$$ $$\frac{a+b}{2}$$ $$Q(\frac{1}{4})$$ where $$Q$$ is the quantile function $$Q(\frac{3}{4})$$ where $$Q$$ is the quantile function 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 $$[\mu - \pi, \mu + \pi]$$ $$f(x) = \frac{1}{2 \pi I_0(\beta)} \exp[\beta \cos(x - \mu)], \; x \in [\mu - \pi, \mu + \pi]$$ $$\mu$$ $$F(x) = \int_{\mu - \pi} ^x f(t) dt, \; x \in [\mu - \pi, \mu + \pi]$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$\mu$$ $$1 - \frac{I_1(\beta)}{I_0(\beta)}$$, where $$I_n$$ is the modified Bessel function of order $$n$$ $$0$$ $$-\beta \frac{I_1(\beta)}{I_0(\beta)} + \ln[2 \pi I_0(\beta)]$$ 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. 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 $$(0, \infty)$$ $$f(x) = \sqrt{\frac{\lambda}{2 \pi x^3}} \exp\left[-\frac{\lambda (x - \mu)^2}{2 \mu^2 x}\right], \; x \in (0, \infty)$$ $$\mu \left[ \sqrt{1 + \left(\frac{3 \mu}{2 \lambda}\right)^2} - \frac{3 \mu}{2 \lambda} \right]$$ $$F(x) = \Phi\left[\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu} - 1\right)\right] + \exp\left(\frac{2 \lambda}{\mu}\right) \Phi\left[-\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu} + 1\right)\right], \; x \in (0, \infty)$$ where $$\Phi$$ is the standard normal distribution function $$Q(p) = F^{-1}(p), \; p \in (0, 1)$$ where $$F$$ is the distribution function $$M(t) = \exp \left[ \frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2 \mu^2}{\lambda} t} \right)\right], \; t \in (-\infty, \frac{\lambda}{2 \mu^2})$$ $$\varphi(t) = \exp \left[ \frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2 \mu^2}{\lambda} i t} \right)\right], \; t \in (-\infty, \infty)$$ $$\mu$$ $$\frac{\mu^3}{\lambda}$$ $$3 \sqrt{\frac{\mu}{\lambda}}$$ $$15 \frac{\mu}{\lambda}$$ $$Q(\frac{1}{2})$$ 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 Wald distribution is named for Abraham Wald. 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 $$f(x) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} \exp\left(-\left(\frac{x}{\lambda}\right)^k\right), \; x \in (0, \infty)$$ $$\lambda \left(\frac{k-1}{k}\right)^{1/k}, \; k \in (1, \infty)$$ $$F(x) = 1 - \exp\left(-\left(\frac{x}{\lambda}\right)^k\right), \; x \in (0, \infty)$$ $$Q(p) = \lambda \left(- \ln(1 - p)\right)^{1/k}, \; p \in (0, 1)$$ $$M(t) = \sum_{n=0}^\infty \frac{t^n \lambda^n}{n!} \Gamma\left(1 + \frac{n}{k}\right), \; t \in (-\infty, \infty, \; k \in (1, \infty)$$ 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. $$\lambda \Gamma\left(1 + \frac{1}{k}\right)$$ where $$\Gamma$$ is the gamma function $$\lambda^2 \left[\Gamma\left(1 + \frac{2}{k}\right) - \Gamma^2\left(1 + \frac{1}{k}\right) \right]$$ where $$\Gamma$$ is the gamma function $$\frac{\Gamma(1 + 3/k) -3 \Gamma(1 + 1/k) \Gamma(1 + 2/k) + 2 \Gamma^3(1 + 1/k)}{[\Gamma(1 + 3/k) - \Gamma^2(1 + 1/k)]^{3/2}}$$ $$\frac{-6 \Gamma^4(1 + 1/k) + 12 \Gamma^2(1 + 1/k) \Gamma(1 + 2/k) -3 \Gamma^2(1 + 2/k) - 4 \Gamma(1 + 1/k)\Gamma(1 + 3/k) + \Gamma(1 + 4/k)}{[\Gamma(1 + 2/k) - \Gamma^2(1 + 1/k)]^2}$$ $$\lambda [\ln(2)]^{1/k}$$ $$\lambda [\ln(4) - \ln(3)]^{1/k}$$ $$\lambda [\ln(4)]^{1/k}$$ 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. zeta distribution Zipf distribution discrete The zeta distribution models ranks and sizes of certain randomly chosen items. $$s \in [1, \infty)$$ $$\{1, 2, \ldots\}$$ $$f(x) = \frac{x^{-s}}{\zeta(s)}, \; x \in \{1, 2, \ldots\}$$ where $$\zeta$$ is the zeta function $$1$$ $$F(x) = \sum_{n=1}^x f(n), \; x \in \{1, 2, \ldots\}$$ where $$f$$ is the probability density function $$Q(p) = F^{-1}(p), \; p \in [0, 1)$$ where $$F$$ is the distribution function $$\frac{\zeta(s - 1)}{\zeta(s)}, \; s \in (2, \infty)$$ $$\frac{\zeta(s - 2)}{\zeta(s)} - \left[\frac{\zeta(s - 1)}{\zeta(s)}\right]^2$$ $$Q(\frac{1}{2})$$ 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 Zipf distribution is named for the American linguist George Kingsley Zipf, who studied the distribution in the context of the frequency of words. 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 folded normal distribution continuous The folded normal distribution governs $$|X|$$ when $$X$$ has a normal distribution $$\mu \in (-\infty, \infty)$$, the location parameter $$\sigma \in (0, \infty$$, the scale parameter $$[0, \infty)$$ $$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \left[ \exp\left(-\frac{(x + \mu)^2}{2 \sigma^2}\right) + \exp \left(-\frac{(x - \mu)^2}{2 \sigma^2}\right) \right], \quad x \in (0, \infty)$$ 0 $$F(x) = \frac{1}{2} \left[ \erf\left(\frac{x + \mu}{\sqrt{2} \sigma} \right) + \erf\left(\frac{x - \mu}{\sqrt{2}\sigma}\right)\right], \quad x \in [0, \infty)$$ where $$\erf$$ is the error 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 $$\mu^2 + \sigma^2 - \left\{\sigma \sqrt{\frac{2}{\pi}} \exp\left(-\frac{\mu^2}{2 \sigma^2} \right) + \mu \left[1 - 2 \Phi(-\frac{\mu}{\sigma}\right]\right\}^2$$ 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 half normal distributon continuous The half normal distribution governs $$|X|$$ when $$X$$ has a normal disstribution with mean 0. $$\sigma \in (0, \infty$$, the scale parameter $$[0, \infty)$$ $$f(x) = \sqrt{\frac{2}{\sigma \pi}} \exp\left(-\frac{x^2}{2 \sigma^2}\right), \quad x \in [0, \infty)$$ 0 $$F(x) = \erf \left(\frac{y}{\sqrt{2} \sigma} \right), \quad x \in [0, \infty)$$, where $$\erf$$ is the error function $$Q(p) = F^{-1}(p)$$ where $$F$$ is the distribution function $$\sigma \sqrt{\frac{2}{\pi}}$$ $$\sigma^2 \left(1 - \frac{2}{\pi}\right)$$ $$\frac{1}{2} \log\left(\frac{\pi \sigma^2}{2}\right) + \frac{1}{2}$$ $$\mu(n) = \frac{\pi^{(n-1)/2}}{\sigma^n} \Gamma\left(\frac{1}{2}(n + 1) \right)$$ where $$\Gamma$$ is the gamma function $$\frac{\sqrt{2}(4 - \pi)}{(\pi - 2)^{3/2}}$$ $$\frac{8(\pi - 3)}{(\pi - 2)^2}$$ $$Q(\frac{1}{2})$$ 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 birthday distribtuion 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 matching distribtuion 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 $$Q(\frac{1}{2})$$ 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 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 $$G(t) = \prod_{i=1}^k \frac{m - (i - 1)}{m - (i - 1)t}, \quad |t| \lt \frac{m}{k - 1}$$ $$\sum_{i=1}^k \frac{m}{m - i + 1}$$ $$\sum_{i=1}^k \frac{(i-1)m}{(m - i + 1)^2}$$ continuous uniform distribution continuous 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. $$A \in \{-\infty , \infty\}$$, left limit $$B \in \{A , \infty\}$$, right limit (B>A) $$\{A, B\}$$ $$\begin{cases} \frac{1}{b - a}, \text{for } x \in [a,b] \\ 0 , \text{otherwise} \end{cases}$$ $$\begin{cases} 0 , \text{for } x \le a \\ \frac{x-a}{b-a} , \text{for } x \in [a,b] \\ 1 , \text{for } x \ge b \end{cases}$$ $$\frac{1}{2}(a+b)$$ $$\frac{1}{2}(a+b)$$ $$\frac{1}{12}(b-a)^2$$ standard continuous uniform distribution standard uniform distribtion continuous The standard continuous uniform distribution is a special case of the (general) continuous uniform distribution defined on [0 , 1]. $$\{0, 1\}$$ $$\begin{cases} \text{for } x \in [0,1] \\ 0 , \text{otherwise} \end{cases}$$ $$\begin{cases} 0 , \text{for } x \le 0 \\ \frac{x-a}{b-a} , \text{for } x \in [0,1] \\ 1 , \text{for } x \ge 1 \end{cases}$$ $$\frac{1}{2}$$ $$\frac{1}{2}$$ $$\frac{1}{12}$$ 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 $$x \in [0; \infty)\!$$ $$\frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!\,}$$ $$\frac{\gamma(k, \lambda x)}{(k-1)!}=1-\sum_{n=0}^{k-1}e^{-\lambda x}(\lambda x)^{n}/n!$$ $$k/\lambda$$ no simple closed form $$k /\lambda^2$$ 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 $$(0, \infty)$$ $$f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}; x \in (0, \infty)$$ Standard Wald 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 $$(0, \infty)$$ $$f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}$$ 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. $$\gamma \in (0, \infty)$$ $$\delta \in (0, \infty)$$ $$\kappa \in (0, \infty)$$ $$(0, \infty)$$ $$f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta(\kappa^x-1)}{log(\kappa)})$$ 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. $$\alpha_1,...,\alpha_n \in (0, \infty), \alpha_i \neq \alpha_j for i \neq j$$ $$[0, \infty)$$ $$f(x) = \sum_{i=1}^{n}(\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j})$$ 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 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. $$\alpha_1,...,\alpha_n \in (0, \infty), \alpha_i \neq \alpha_j for i \neq j$$ $$p_i > 0, \sum_{i=1}^{n} p_i = 1$$ $$(0, \infty)$$ $$f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}$$ 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 $$(0, \infty)$$ $$f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}$$ 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 $$(-\infty, \infty)$$ $$f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}$$ Minimax distribution continuous The Minimax distribution is an alternative two-parameter distribution of the Beta distribution. $$\beta \in (0, \infty)$$ $$\gamma \in (0, \infty)$$ $$(0, 1)$$ $$f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}$$ 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 $$\delta \in (0, \infty)$$ $$\kappa \in (0, \infty)$$ $$\gamma \in [0, \infty)$$ $$(0, \infty)$$ $$f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}$$ Benford's law first 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. $$d \in {1,...,9}$$ $$b = 10$$, the log base $$(0, 1)$$ $$P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d})$$ Standard Triangular distribution continuous The standard triangular distribution is a special case of the triangular distribution where $$a=-1, b=1, m=0$$. none $$[-1, 1]$$ $$f(x) = \begin{cases} x+1, -1\lt x\lt 0 \\ 1-x, 0 \leq x\lt 1 \end{cases}$$ 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. $$\delta \in (0, \infty)$$ $$\gamma \in (0, \infty)$$ $$(0, \infty)$$ $$f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}$$ Wigner semicircle distribution Half Circle distribution continuous The Wigner semicircle distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity. $$r \in (0, \infty)$$, the radius $$[-r, r]$$ $$f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r]$$ 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. $$n \in (0, \infty)$$ $$a \in (-\infty, \infty)$$ $$b \in (a, \infty)$$ $$m=(b-a)\theta+a$$ $$(a, b)$$ $$f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a\lt x\le m \\ \frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x\lt b \end{cases}$$ Extreme Value Type I 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. $$\alpha \in (0, \infty)$$ $$\beta \in (0, \infty)$$ $$(-\infty, \infty)$$ $$f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}$$ 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 $$[0, \infty)$$ $$f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}$$ 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 $$(0, \infty)$$ $$f(x)=\frac{1}{\sigma}(1+\kappa \frac{(x-\mu)}{\sigma})^{-1-\frac{1}{\kappa}}$$ Kolmogorov-Smirnov test Kolmogorov distribution continuous The Kolmogorov-Smirnov test can be modified to serve as a goodness of fit test. none $$(0, \infty)$$ $$f(x)=1-2[\exp{-x^2}-\exp{-4 x^2}+\exp{-9 x^2}-\exp{-16 x^2}+...]$$ Logistic-Exponential continuous infinte $$\alpha = \beta = \frac{1}{2}$$ Power-Function continuous nonsymmetric finite $$f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha}$$ Student's T Non-Central continuous nonsymmetric infinite $$f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx$$ Inverted Gamma continuous nonsymmetric finite positive $$\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)$$ Fisher-Tippett continuous nonsymmetric finite positive $$\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)$$ Gibrat's continuous nonsymmetric finite positive $$\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]$$ Gompertz continuous nonsymmetric finite positive $$b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]$$ MultiNomial discrete nonsymmetric finite positive $$f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}$$ NegativeMultiNomial discrete nonsymmetric finite positive $$f(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}}$$ Negative Hypergeometric 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 $$B$$. $$W \in \{1,2,...\}$$ $$B \in \{1,2,...\}$$ $$b \in \{1,2,...,B\}$$ $$x=\{0,1,...,W\}$$ $$f(x) \frac{ { x+b-1 \choose x} {W+B-b-x \choose W-x} }{ {W+B \choose W} }$$ Power Series discrete infinite positive $$f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!$$ Beta-Pascal discrete infinite positive $$f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!$$ Gamma-Poisson discrete infinite positive $$f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!$$ Polya discrete finite positive $$f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!$$ Gamma-Normal bivariate continuous infinite $$f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!$$ Discrete Weibull discrete infinite positive $$f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!$$ Noncentral Beta continuous positive $$f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1) \!$$ Arctangent continuous infinite positive $$f(x; \lambda, \phi)= \frac{\lambda}{[\arctan(\lambda \phi)+\pi/2] [1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty \lt \lambda \lt \infty) \!$$ Log Gamma continuous infinite $$f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}$$, where $$(-\infty \lt x \lt \infty) \!$$ 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 http://www.jstor.org/stable/2283941 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 http://www.jstor.org/stable/2283679 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 http://www.jstor.org/stable/2699817 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 10.1016/j.aim.2004.06.011 beta distribution arcsine distribution The beta distribution with parameters $$\alpha = \frac{1}{2}$$ and $$\beta = \frac{1}{2}$$ is the arcsine distribution. special case 10.1112/plms/s3-65.2.326 beta distribution standard uniform distribution The beta distribution with parameters $$\alpha = 1$$ and $$\beta = 1$$ is the standard uniform distribution. special case 10.1080/0266476042000214501 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 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 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 %beta to beta 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 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 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 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 10.1016/j.compedu.2006.06.003 binomial distribution Bernoulli distribution The binomial distribution with parameters $$n = 1$$ and $$p \in [0, 1]$$ is the Bernoulli distribution with parameter $$p$$. special case 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 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 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 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 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 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 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 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 chi-square distribution standard 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 10.1016/j.compedu.2006.06.003 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 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 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 chi-square distribution standard normal distribution Student's 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 Student's's $$t$$-distribution with $$n$$ degrees of freedom. transformation 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 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 continuous uniform distribution standard uniform distribution The continuous uniform distribution on $$[0, 1]$$ is the standard uniform distribution special case 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 standard uniform distribtion 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 standard 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 standard 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 standard uniform distribution standard Cauchy distribution If $$X$$ has the standard uniform distribution then $$\tan[\pi(X - \frac{1}{2})]$$ has the standard Cauchy distribution. nonlinear transformation standard uniform distribution arcsine distribution If $$X$$ has the standard uniform distribution then $$\sin^2(\frac{\pi}{2} X)$$ has the arcsine distribution. nonlinear transformation standard 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 standard 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 standard 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 standard 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 standard 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 standard 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 standard 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 standard 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 standard 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 standard 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 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 exponential distribution standard 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 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 $$\frac{1}{\lambda}$$. convolution 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 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 exponential distribution Gumbel 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 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 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 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 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 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 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 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 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 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 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 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 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 gamma distribution standard 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 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 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 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 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 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 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 Gumbel distribution Gumbel 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 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 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 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 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 hyperbolic secant distribution standard 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 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 Irwin-Hall distribution standard uniform distribution The Irwin-Hall distribution with parameter $$1$$ is the standard uniform distribution. special case 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 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 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 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 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 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 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 %logistic to standard uniform logistic distribution standard 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 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 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 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 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 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 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 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 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 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 $$p$$. special case 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 negative binomial distribution standard 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 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 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 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 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 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 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 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 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 normal distribution standard normal If $$X$$ has the normal distribution with mean $$\mu = 0$$ and standard deviation $$\sigma = 1$$, then $$X$$ has a standard normal distribution. special case standard 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 normal distribution Student 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 Student's t distribution with $$n-1$$ degrees of freedom. transformation 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 standard normal distribution standard 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 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 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 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 Pareto distribution standard 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 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 Poisson distribution standard 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 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 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 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 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 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 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 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 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 Rayleigh distribution Standard 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 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 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 semicircle distribution standard 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 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. 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. 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. 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 %Student's t to F Student's t-distribution F-distribution if $$X$$ has the Student's 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 Student t-distribution Cauchy distribution The Student's t-distribution with 1 degree of freedom is the standard Cauchy distribuiton. special case U-quadratic distribution standard 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 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 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 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 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 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 General Normal distribution Standard Normal distribution The general normal distribution with $$\mu=0$$ and $$\sigma^2=1$$ is called the standard normal transformation Student's t distribution Standard Normal distribution As $$n\longrightarrow\infty$$, the t-distribution approaches the normal distribution with mean 0 and variance 1 limiting Fisher's F distribution Student's t distribution The square root of a Fisher's F distribution is a Student's t distribution transformation Binomial distribution General 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 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 Noncentral Student's 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 Standard 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 Standard 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 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 Fisher-Tippett distribution Gumbel distribution The Gumbel distribution is a particular case of the Fisher-Tippett distribution where $$\mu=0, \beta=1$$ special case Log-Normal distribution Gibrat's distribution Gibrat's law is a special case of the log-normal distribution where special case $$\mu=0, x=1$$ Standard Cauchy distribution Cauchy distribution If $$X$$ is a standard Cauchy distribution, then $$Y = x_0 + \gamma X$$ is a Cauchy distribution transformation Multinomial distribution Binomial distribution When $$k=2$$, the multinomial distribution is the binomial distribution transformation 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 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 Rectangular distribution Beta-Binomial distribution For $$a=b=1)$$, the rectangular distribution reduces to the beta-binomial distribution as a special case special case Zipf distribution Zeta distribution The zeta distribution is equivalent to the Zipf distribution for infinite N. limiting Power Series distribution Poisson distribution The power series$$(c, A(c))$$ distribution becomes a Poisson distribution when $$\mu=c, A(c)=e^c$$ transformation 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 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 Pascal distribution Geometric distribution The geometric distribution is a special case of the Pascal distribution where $$n=1$$ special case Discrete Weibull distribution Geometric distribution The geometric distribution is a particular case of the discrete Weibull distribution where $$\beta=1$$ special case 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 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 Inverse Gaussian distribution Standard Normal distribution As $$\lambda \to \infty$$, the inverse Gaussian distribution becomes more like a standard normal (Gaussian) distribution limiting 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 Generalized Gamma distribution Gamma distribution The gamma distribution is a special case of the generalized gamma distribution where $$\gamma=1$$ special case Inverse Gaussian 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 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 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 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 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 Makeham distribution Gompertz distribution The Gompertz distribution is a special case of the Makeham distribution where $$\gamma=0$$ special case 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 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 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 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 Standard 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 Laplace distribution Error distribution The error distribution is a special case of Laplace distribution where $$\alpha_1=\alpha_2$$ special case Standard Uniform distribution Standard 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 Standard Uniform distribution Standard Power distribution If X is and independent random variable with standard uniform distribution, then $$X^{1/\beta}$$ is a standard power distribution transformation Standard Uniform distribution Standard Power distribution If $$X$$ is a standard uniform distribution, then $$X_(n)$$ is a standard power distribution special case IDB distribution Rayleigh distribution The Rayleigh distribution is a special case of the IDB distribution where $$\delta=2/\alpha, \gamma=0$$ special case, transformation Weibull distribution Rayleigh distribution The Rayleigh distribution is a special case of the Weibull distribution where $$\beta=1$$ special case Triangular distribution Standard Triangular distribution The standard triangular distribution is a special case of the triangular distribution where $$a=-1, b=1, m=0$$. special case Log-Logistic distribution Lomax distribution The Lomax distribution is a special case of the Log-logistic distribution where $$\kappa = 1$$ special case 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 Exponential distribution Erlang distribution noncentral Student's t-distribution Student's t-distribution logistic exponential distribution exponential distribution standard uniform distribution benford distribution gamma distribution inverted gamma distribution Cauchy distribution standard Cauchy distribution standard Cauchy distribution hyperbolic secant distribution negative binomial distribution negative multinomial distribuion gamma distribution pascal distribution Discrete Uniform Distribution Pascal Distribution power-series distribuion logarithmic distribution pascal distribuion beta pascal distribution polya distribuion binomial distribution geometric distribuion pascal distribution pascal distribuion normal distribuion beta distribution normal distribution generalized gamma distribution log-normal distribution inverse gaussian distribution Wald distribution chi-square distribution chi distribution chi-square distribution exponential distribution beta distribution inverted beta distribution hypoexponential distribution Erlang distribution doubly noncentral t distribution noncentral t distribution noncentral f distribution f distribution hyperexponential distribution exponential distribution exponential distribution Rayleigh distribution standard uniform distribution gompertz distribution error distributon Laplace distribution standard uniform distribution uniform distribution standard power distribution standard uniform distribution minimax distribution standard power distribution power distribution standard power distribution generalized Pareto distribution Pareto distribution Weibull distribution Extreme-value distribution lomax distribution log-logistic distribution TSP distribution triangular distribution