\(I_A(x)=\begin{cases} 1, & x\in A\\ 0, & x\not\in A \end{cases}\)
\(\Gamma(x)=\int_0^{+\infty}{t^{x-1}e^{-t}\,dt},\,x>0\)
\(\Gamma(x+1)=x\Gamma(x),\,x>0\quad \Gamma(n)=(n-1)!,\,n\in\mathbb{N}\)
\(B(x,\,y)=\int_0^1{t^{x-1}(1-t)^{y-1}\,dt},\,x,\,y>0\)
\(B(x,\,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)
\(Uniform\left(\{a,\ldots,a+n-1\}\right)\quad a\in\mathbb{Z},\,n\in\mathbb{N}\)
\(f(x)=\dfrac{1}{n}I_S(x)\quad S=\{a,\ldots,a+n-1\}\)
\(E[X]=a+\dfrac{n-1}{2}\quad Var[X]=\dfrac{n^2-1}{12}\)
\(M(t)=\dfrac{e^{at}\left(1-e^{nt}\right)}{n(1-e^t)}\)
\(Binomial(n,\theta)\quad n\in\mathbb{N}\quad \theta\in]0,1[\)
\(f(x)=\binom{n}{x} \theta^x (1-\theta)^{n-x}I_S(x)\quad S=\{0,\ldots,n\}\)
\(E[X]=n\theta\quad Var[X]=n\theta(1-\theta)\)
\(M(t)=\left(\theta e^t+(1-\theta)\right)^n\)
\(Hypergeometric(N,M,n)\)
\(N\in\mathbb{N}\quad M\in\{1,\ldots,N\}\quad n\in\{1,\ldots,N\}\)
\(f(x)=\dfrac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}}I_S(x)\)
\(S=\left\{\max\{0,n-N+M\},\ldots,\min\{n,M\}\right\}\)
\(E[X]=n\dfrac{M}{N}\quad Var[X]=n\dfrac{M}{N}\dfrac{N-M}{N}\dfrac{N-n}{N-1}\)
\(NegativeBinomial(r,\theta)\quad r\in\mathbb{N}\quad \theta\in]0,1[\)
\(f(x)=\binom{x-1}{r-1} \theta^r(1-\theta)^{x-r}I_S(x)\quad S=\{r,r+1,\ldots\}\)
\(E[X]=\dfrac{r}{\theta}\quad Var[X]=\dfrac{r(1-\theta)}{\theta^2}\)
\(M(t)=\left(\dfrac{\theta e^t}{1-(1-\theta)e^t}\right)^r\)
\(Geometric(\theta)\equiv NegativeBinomial(1,\theta)\)
\(Poisson(\lambda)\quad \lambda\in \mathbb{R}^+\)
\(f(x)=e^{-\lambda}\dfrac{\lambda^{x}}{x!}I_S(x)\quad S=\mathbb{N}_0\)
\(E[X]=Var[X]=\lambda\)
\(M(t)=e^{\lambda\left(e^t-1\right)}\)
\(Uniform(\alpha,\beta)\quad \alpha,\,\beta\in \mathbb{R},\;\alpha<\beta\)
\(f(x)=\dfrac{1}{\beta-\alpha}I_S(x)\quad S=[\alpha,\beta]\)
\(E[X]=\dfrac{\alpha+\beta}{2}\quad Var[X]=\dfrac{(\beta-\alpha)^2}{12}\)
\(M(t)=\dfrac{e^{\beta t}-e^{\alpha t}}{(\beta-\alpha)t},\ t\neq 0\)
\(Beta(\alpha,\beta)\quad \alpha,\,\beta\in\mathbb{R}^+\)
\(f(x)=\dfrac{1}{B(\alpha,\,\beta)}x^{\alpha-1}(1-x)^{\beta-1}I_S(x)\quad S=[0,1]\)
\(E[X]=\dfrac{\alpha}{\alpha+\beta}\quad Var[X]=\dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)
\(Gamma(\alpha,\beta)\quad \alpha,\,\beta\in\mathbb{R}^+\)
\(f(x)=\dfrac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}I_S(x)\quad S=\mathbb{R}_0^+\)
\(E[X]=\dfrac{\alpha}{\beta}\quad Var[X]=\dfrac{\alpha}{\beta^{2}}\)
\(M(t)=\left(\dfrac{\beta}{\beta-t}\right)^\alpha, \; t<\beta\)
\(\chi_{(n)}^2\equiv Gamma\left(\frac{n}{2},\frac{1}{2}\right)\quad n\in\mathbb{N}\)
\(Exponential(\lambda)\equiv Gamma(1,\lambda)\)
\(Normal(\mu,\sigma^2)\quad \mu\in\mathbb{R},\;\sigma^2\in\mathbb{R}^+\)
\(f(x)=\dfrac{1}{\sqrt{2\pi\sigma^2}} e^{-\dfrac{(x-\mu)^2}{2\sigma^2}}\)
\(E[X]=\mu\quad Var[X]=\sigma^2\)
\(M(t)=e^{\mu t +\frac{\sigma^2}{2}t^2}\)
\(Lognormal(\mu,\sigma^2)\quad \mu\in\mathbb{R},\;\sigma^2\in\mathbb{R}^+\)
\(f(x)=\dfrac{1}{x\sqrt{2\pi\sigma^2}} e^{-\dfrac{(\log x-\mu)^2}{2\sigma^2}}I_S(x)\quad S=\mathbb{R}^{+}\)
\(E[X]=e^{\mu+\frac{\sigma^2}{2}}\quad Var[X]=\left(e^{\sigma^2}-1\right)e^{2\mu+\sigma^2}\)
\(E[X^k]=e^{\mu k+\frac{\sigma^2}{2}k^2}\)
\(Weibull(\alpha,\beta)\quad \alpha,\,\beta\in\mathbb{R}^+\)
\(f(x)=\dfrac{\beta}{\alpha}\left(\dfrac{x}{\alpha}\right)^{\beta-1}e^{-\left(\frac{x}{\alpha}\right)^\beta}I_S(x)\quad S=\mathbb{R}_0^{+}\)
\(E[X]=\alpha\Gamma\left(1+\frac{1}{\beta}\right)\quad Var[X]=\alpha^2\left(\Gamma\left(1+\frac{2}{\beta}\right)-\Gamma\left(1+\frac{1}{\beta}\right)^2\right)\)
\(E[X^k]=\beta^{\frac{k}{\alpha}}\Gamma\left(1+\frac{k}{\alpha}\right)\)