gamma distribution

$$f_{alpha, beta}(x)=left{begin{array}{ll}{frac{1}{Gamma(alpha) beta^{alpha}} x^{alpha-1} e^{-x / beta},} & {x>0} \ {0,} & {x leq 0}end{array}right. qquad alpha>0, beta>0$$

Here,

$$Gamma(alpha)=int_{0}^{infty} z^{alpha-1} e^{-z} d z, quad alpha>0$$

is the Euler gamma function, $n !=Gamma(n+1)$

$$E[X] = alpha beta$$ $$text{Var}[X] = alphabeta^2$$ $${m_{X} :(-infty, 1 / beta) rightarrow mathbb{R}, quad m_{X}(t)=(1-beta t)^{-alpha}}$$