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

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 $\Gamma: [0,\infty] \to R$ is the gamma function, defined as $\Gamma(t) = \int_{0}^{\infty}{x^{t-1} e^{-x} dx}$ .
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 $n, \lambda$ .