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 chisquared random variable.

2.
The function $\mathrm{\Gamma}:[0,\mathrm{\infty}]\to R$ is the gamma function, defined as $\mathrm{\Gamma}(t)={\int}_{0}^{\mathrm{\infty}}{x}^{t1}{e}^{x}\mathit{d}x$.

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 $.
Title  gamma random variable 

Canonical name  GammaRandomVariable 
Date of creation  20130322 11:54:27 
Last modified on  20130322 11:54:27 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  14 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 6000 
Classification  msc 6200 
Synonym  gamma distribution 
Defines  Erlang random variable 