## Gamma Distribution

### A continuous random variable $X$ is said to have an gamma distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by

`$$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{\alpha^\beta}{\Gamma(\beta)}x^{\beta -1}e^{-\alpha x}, & \hbox{$x>0;\alpha, \beta >0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$`

## Mean of Gamma Distribution

The mean or expected value of gamma random variable is

### $E(X)= \dfrac{\beta}{\alpha}$

## Variance of Gamma distribution

The variance of gamma random variable is

### $V(X) = \dfrac{\beta}{\alpha^2}$.

## Harmonic Mean of Gamma Distribution

The harmonic mean of gamma random variable is

### $H=\frac{\beta-1}{\alpha}$.

## Mode of Gamma distribution

The mode of gamma random variable is

### $\dfrac{\beta-1}{\alpha}$.

## M.G.F. of Gamma Distribution

The moment generating function of gamma random variable is