gamma random variable

A gamma random variable with parameters $\alpha >0$ and $\lambda >0$ is one whose probability density function is given by

$f_X(x)={\displaystyle \frac{{\lambda }^{\alpha }}{\mathrm{\Gamma }(\alpha )}}{x}^{\alpha -1}{e}^{-\lambda x}$

for $x>0$, and is denoted by $X\sim Gamma(\alpha ,\lambda )$.

Notes:

1. Gamma random variables are widely used in many applications. Taking $\alpha =1$ reduces the form to that of an exponential random variable. If $\alpha =\frac{n}{2}$ and $\lambda =\frac{1}{2}$, this is a chi-squared random variable.

2. The function $\mathrm{\Gamma }:[0,\mathrm{\infty }]\to R$ is the gamma function, defined as $\mathrm{\Gamma }(t)={\int }_0^{\mathrm{\infty }}{x}^{t-1}{e}^{-x}𝑑x$.

3. The expected value of a gamma random variable is given by $E[X]=\frac{\alpha }{\lambda }$, and the variance by $Var[X]=\frac{\alpha }{{\lambda }^2}$

4. The moment generating function of a gamma random variable is given by $M_X(t)={(\frac{\lambda }{\lambda -t})}^{\alpha }$.

If the first parameter is a positive integer, the variate is usually called Erlang random variate. The sum of $n$ exponentially distributed variables with parameter $\lambda$ is a gamma (Erlang) variate with parameters $n,\lambda$.