Definition

A continuous RV $X$ is called gamma with parameters $\alpha>0$ and, $\lambda>0$ if it has pdf f(x)=\left\{ \begin{array} 1\frac{\lambda^\alpha}{\Gamma(\alpha)}x^{r-1}e^{-\lambda x}&x>0\\ 0&x\le0 \end{array} \right. where $$\Gamma(\alpha)=\int_{0}^{\infty} y^{\alpha-1}e^{-y}dy$$where $$\begin{align*} E[X]&=\frac{\alpha}{\lambda}&\\\\ E[X^{k}]&=\frac{1}{\lambda^{k}}(k+\alpha-1)(k+\alpha-2)\cdots \alpha\\\\ \mbox{Var}(X)&=\frac{\alpha}{\lambda^{2}} \end{align*}$$ # Theorem (Properties of Gamma Function) 1. $\Gamma(1)=1$ 2. $\Gamma(1/2)=\sqrt{\pi}$ 3. Recursive Property: $$\Gamma(\alpha)=(r-1)\Gamma(r-1)=(r-1)!$$