distribution parameters support density f(x) mean variance excel function matlab/octave r numpy/scipy c++
binomial B(n,p) {0,1,...,n} [[$ \frac{n!}{x!(n-x)!} p^x(1-p)^{n-x} $]] np np(1-p) BINOMDIST(x,n,p,FALSE)
BINOMDIST(x,n,p,TRUE)
BINOM.INV(n, p, α)
BINOM.INV(n, p, RAND())		 binopdf(x, n, p)
binocdf(x, n, p)
binoinv(y, n, p)
binornd(n, p)		 dbinom(x, n, p)
pbinom(x, n, p)
qbinom(y, n, p)
rbinom(1, n, p)		 stats.binom.pmf(x, n, p)
stats.binom.cdf(x, n, p)
stats.binom.ppf(y, n, p)
stats.binom.rvs(n, p)		 #include <random

default_random_engine dre;
binomial_distribution<int> bd(n, p);
int m = bd(dre);
poisson Pois(λ) {0,1,2,...} [[$ \frac{\mu^x e^{-\mu}}{x!} $]] λ λ POISSON(x, λ, FALSE)
POISSON(x, λ, TRUE)
none
none		 poisspdf(x, lambda)
poisscdf(x, lambda)
poissinv(y, lambda)
poissrnd(lambda)		 dpois(x, lambda)
ppois(x, lambda)
qpois(y, lambda)
rpois(1, lambda)		 stats.poisson.pmf(x, lambda)
stats.poisson.cdf(x, lambda)
stats.poisson.ppf(y, lambda)
stats.poisson.rvs(lambda, size=1)		 #include <random

default_random_engine dre;
poisson_distribution<int> pd(lambda);
int m = pd(dre);
normal N(μ, σ) (-∞,∞) [[$ \frac{1}{\sqrt{2\pi \sigma}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $]] μ σ2 NORMDIST(x, μ, σ, FALSE)
NORMDIST(x, μ, σ, TRUE)
NORMINV(α, μ, σ)
NORMINV(RAND(), μ, σ)		 normpdf(x, mu, sigma)
normcdf(x, mu, sigma)
norminv(y, mu, sigma)
normrnd(mu, sigma)		 dnorm(x, mu, sigma)
pnorm(x, mu, sigma)
qnorm(y, mu, sigma)
rnorm(1, mu, sigma)		 stats.norm.pdf(x, mu, sigma)
stats.norm.cdf(x, mu, sigma)
stats.norm.ppf(y, mu, sigma)
stats.norm.rvs(mu, sigma)		 #include <random

default_random_engine dre;
normal_distribution<double> nd(mu, sigma);
double x = nd(dre);
gamma Γ(k, θ) [0,∞) [[$ x^{k-1}\frac{exp(\frac{-x}{\theta})}{\Gamma(k) \theta^k} $]] kθ kθ2 GAMMADIST(x, k, θ, FALSE)
GAMMADIST(x, k, θ, TRUE)
GAMMAINV(α, k, θ)
GAMMAINV(RAND(), k, θ)		 gampdf(x, k, theta)
gamcdf(x, k, theta)
gaminv(y, k, theta)
gamrnd(k, theta)		 dgamma(x, k, scale=theta)
pgamma(x, k, scale=theta)
qgamma(y, k, scale=theta)
rgamma(1, k, scale=theta)		 stats.gamma.pdf(x, k, scale=theta)
stats.gamma.cdf(x, k, scale=theta)
stats.gamma.ppf(y, k, scale=theta)
stats.gamma.rvs(k, scale=theta)		 #include <random

default_random_engine dre;
gamma_distribution<double> gd(k, theta);
double x = gd(dre);
exponential Exp(λ) [0, ∞) [[$ \lambda e^{-\lambda x} $]] λ-1 λ-2 EXPON.DIST(x, λ, FALSE)
EXPON.DIST(x, λ, TRUE)
GAMMAINV(y, 1, 1/λ)
GAMMAINV(RAND(), 1, 1/λ)		 exppdf(x, lambda)
expcdf(x, lambda)
expinv(y, lambda)
exprnd(lambda)		 dexp(x, lambda)
pexp(x, lambda)
qexp(y, lambda)
rexp(1, lambda)		 stats.expon.pdf(x, scale=1.0/lambda)
stats.expon.cdf(x, scale=1.0/lambda)
stats.expon.ppf(x, scale=1.0/lambda)
stats.expon.rvs(scale=1.0/lambda)		 #include <random

default_random_engine dre;
exponential_distribution<double> ed(lambda);
double x = ed(dre);
chi-squared Χ^^2^^(ν) [0, ∞) [[$ \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} e^{-x/2} $]] ν 2ν CHISQ.DIST(x, ν, FALSE)
CHISQ.DIST(x, ν, TRUE)
CHISQ.INV(y, ν)
CHISQ.INV(RAND(), ν)		 chi2pdf(x, nu)
chi2cdf(x, nu)
chi2inv(y, nu)
chi2rnd(nu)		 dchisq(x, nu)
pchisq(x, nu)
qchisq(y, nu)
rchisq(1, nu)		 stats.chi2.pdf(x, nu)
stats.chi2.cdf(x, nu)
stats.chi2.ppf(y, nu)
stats.chi2.rvs(nu)		 #include <random

default_random_engine dre;
chi_squared_distribution<double> csd(nu);
double x = csd(dre);
beta Be(α, β) [0, 1] [[$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)}$]] [[$ \frac{\alpha}{\alpha + \beta} $]] [[$ \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} $]] BETADIST(x, α, β, FALSE)
BETADIST(x, α, β, TRUE)
BETAINV(p, α, β)
BETAINV(RAND(), α, β)		 betapdf(x, alpha, beta)
betacdf(x, alpha, beta)
betainvf(y, alpha, beta)
betarnd(alpha, beta)		 dbeta(x, alpha, beta)
pbeta(x, alpha, beta)
qbeta(y, alpha, beta)
rbeta(1, alpha, beta)		 stats.beta.pdf(x, alpha, beta)
stats.beta.cdf(x, alpha, beta)
stats.beta.ppf(y, alpha, beta)
stats.beta.rvs(alpha, beta)		 none
uniform U(a, b) [a, b] [[$ \frac{1}{b-a} $]] [[$ \frac{a+b}{2} $]] [[$ \frac{(b-a)^2}{12} $]] 1/(b-a)
(x-a)/(b-a)
α * (b-a) + a
RAND()*(b-a) + a		 unifpdf(x, a, b)
unifcdf(x, a, b)
unifinv(y, a, b)
unifrnd(a, b)		 dunif(x, a, b)
punif(x, a, b)
qunif(y, a, b)
runif(1, a, b)		 stats.uniform.pdf(x, a, b)
stats.uniform.cdf(x, a, b)
stats.uniform.ppf(y, a, b)
stats.unifrom.rvs(a, b)		 #include <random

default_random_engine dre;
uniform_real_distribution<double> urd(a, b);
double x = urd(dre);
Student's t t(ν) (-∞,∞) [[$ \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu \pi} \Gamma(\frac{\nu}{2})} (1 + \frac{x^2}{\nu})^{-\frac{\nu+1}{2}}$]] [[$ \begin{cases} 0 & \nu > 1 \\ \text{undefined} & \text{otherwise} \end{cases} $]] [[$ \begin{cases} \frac{\nu}{\nu - 2} & \nu > 2 \\ \infty & 1 < \nu \le 2 \\ \text{undefined} & \text{otherwise} \end{cases} $]] T.DIST(x, ν, FALSE)
T.DIST(x, ν, TRUE)
T.INV(α, ν)
T.INV(RAND(), ν)		 dt(x, nu)
pt(x, nu)
qt(y, nu)
rt(1, nu)		 stats.t.pdf(x, nu)
stats.t.cdf(x, nu)
stats.t.ppf(y, nu)
stats.t.rvs(nu)		 #include <random

default_random_engine dre;
student_t_distribution<double> td(nu);
double x = td(dre);
Snedecor's F F(d1, d2) [0, ∞) [[$ \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1+d_2}}}}{x B(d_1, d_2)} $]] [[$ \frac{d_2}{d_2 - 2} $]] //for d,,2,, > 2//// F.DIST(x, d1, d2, FALSE)
F.DIST(x, d1, d2, TRUE)
F.INV(α, d1, d2)
F.INV(RAND(), d1, d2)		 df(x, d1, d2)
pf(x, d1, d2)
qf(y, d1, d2)
rf(1, d1, d2)		 stats.f.pdf(x, d1, d2)
stats.f.cdf(x, d1, d2)
stats.f.ppf(y, d1, d2)
stats.f.rvs(d1, d2)		 #include <random

default_random_engine dre;
fisher_f_distribution<double> fd(d1, d2);
double x = urd(dre);
________ __________________________ _________________________ _________________________ ____________________________________ ____________________________________________

Conjugate Priors

distribution conjugate prior posterior hyperparameters posterior predictive
bernoulli beta [[$ α' = α + \sum_{i=1}^n x_i $]]
[[$ β' = β + n - \sum_{i=1}^n x_i $]]
binomial beta [[$ α' = α + \sum_{i=1}^n x_i $]]
[[$ β' = β + \sum_{i=1}^n N_i - \sum_{i=1}^n x_i $]]
negative binomial (known number of failures) beta
geometric beta
categorical dirichlet
multinomial dirichlet
poisson gamma
hypergeometric (known population N) beta-binomial
distribution conjugate prior posterior hyperparameters posterior predictive
normal (known variance) normal
normal (known mean) inverse gamma
uniform pareto
exponential gamma
gamma (known shape α) gamma

Continuous Distributions by Thickness of Tail

  • Cauchy
  • Student's t
  • Laplace
  • Gaussian
  • sub-Gaussian