Gamma Distribution

Statements

Parameters

• $\alpha >0$: Number of occurrence of a Poisson distribution event.
• $\beta >0$: Average rate of occurrence.

A random variable is said to be Gamma distributed if it is described by the following

Probability Density Function

$$\begin{array}{rrcl}{p}_X(\cdot ,\alpha ,\beta ):& \mathbb{R}& \to & \mathbb{R}\\ & x& \to & \{\begin{array}{rcl}\frac{{\beta }^{\alpha }}{\Gamma (\alpha )}{x}^{\alpha -1}{e}^{-\beta x},& \text{if}& x>0\\ 0,& \text{if}& x\le 0.\end{array}\end{array}$$

Moments

• Expected value: $E[X]=\alpha /\beta$,
• Variance: $\text{Var}(X)=\alpha /{\beta }^2$.

Relationship with Other Distributions

Exponential Distribution

When $\alpha =1$, the Gamma distribution reduces to the exponential distribution with parameter $\beta =\lambda$.

Chi-Square

For a value $m>0$, let $\alpha =m/2$ and $\beta =1/2$. Then, $X$ has a chi-square distribution with $m$ degrees of freedom.