Gamma Distribution

Definition

The Gamma distribution is a generalization of the Exponential Distribution. While an Exponential r.v. represents the waiting time for the first success under conditions of memoryless-ness, we shall see that Gamma r.v. represents the total waiting time for multiple successes

An r.v. $Y$ is said to have the Gamma Distribution with parameters $a$ and $\lambda$, where $a>0$ and $\lambda >0$, if its PDF is

$$f(y)=\frac{1}{\Gamma (a)}(\lambda y{)}^a{e}^{-\lambda y}\frac{1}{y},y>0$$

where $\Gamma (\cdot )$ is the Gamma Function. And we write $Y\sim \mathrm{Gamma}(a,\lambda )$

The expectation of Gamma distribution if $\mathrm{E}[Y]=\frac{a}{\lambda }$

Relationship with Other Distributions

Exponential Distribution

If $X_1,X_2,\dots ,X_n$ i.i.d. $\sim \mathrm{Expo}(\lambda )$, then we will have

$$X_1+X_2+\cdots +X_n\sim \mathrm{Gamma}(n,\lambda )$$

Chi-Squared Distribution

If $X\sim \mathrm{Gamma}(a,\lambda )$, then

$$Y=2\lambda X\sim {\chi }_{2a}^2$$

where ${\chi }_{2a}^2$ denotes the Chi-Squared Distribution with $2a$ degrees of freedom