# gamma random variable

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

 $\displaystyle f_{X}(x)=\frac{\lambda^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-% \lambda x}$

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

Notes:

1. 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. 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. 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. 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 $n,\lambda$.

Title gamma random variable GammaRandomVariable 2013-03-22 11:54:27 2013-03-22 11:54:27 mathcam (2727) mathcam (2727) 14 mathcam (2727) Definition msc 60-00 msc 62-00 gamma distribution Erlang random variable