## 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