Distributions
distributionparameterssupportdensity f(x)meanvarianceexcel functionmatlab/octavernumpy/scipyc++
binomialB(n,p){0,1,…,n} $$\frac{n!}{x!(n-x)!} p^x(1-p)^{n-x}$$ npnp(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);
poissonPois(λ){0,1,2,…} $$\frac{\mu^x e^{-\mu}}{x!}$$ λλPOISSON(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);
normalN(μ, σ)(-,) $$\frac{1}{\sqrt{2\pi \sigma}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$ μσ2NORMDIST(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);
gammaΓ(k, θ)[0,) $$x^{k-1}\frac{exp(\frac{-x}{\theta})}{\Gamma(k) \theta^k}$$ kθkθ2GAMMADIST(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);
exponentialExp(λ)[0, ) $$\lambda e^{-\lambda x}$$ λ-1λ-2EXPON.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-squaredΧ2(ν)[0, ) $$\frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}$$ ν2νCHISQ.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);
betaBe(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
uniformU(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 tt(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 FF(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 d2 > 2F.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);
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