Gamma Distribution

A continuous Random Variable is of gamma distribution if

• Parameters
  • $\alpha ,\lambda >0$
• PDF
  • $f(x)=\{\begin{array}{ll}\frac{\lambda e^{-\lambda x}(\lambda x{)}^{\alpha -1}}{\Gamma (\alpha )},& x\ge 0,\\ 0,& x<0\end{array}$
  • where $\Gamma$ is the Gamma Function
    • $\Gamma (\alpha )={\int }_0^{\infty }{e}^{-y}{y}^{\alpha -1}dy$
    • $\Gamma (1)=1$
    • $\Gamma (\alpha )=(\alpha -1)\Gamma (\alpha -1)$
    • $\Gamma (n)=(n-1)!$
• Mean
  • $\alpha /\lambda$
• Variance
  • $\alpha /{\lambda }^2$
• MGF
  • ${\left(\frac{\lambda }{\lambda -t}\right)}^{\alpha },t<\lambda$