gamma random variable

A gamma random variable with parameters α>0 and λ>0 is one whose probability density functionMathworldPlanetmath is given by


for x>0, and is denoted by XGamma(α,λ).


  1. 1.

    Gamma random variables are widely used in many applications. Taking α=1 reduces the form to that of an exponential random variable. If α=n2 and λ=12, this is a chi-squared random variable.

  2. 2.

    The function Γ:[0,]R is the gamma function, defined as Γ(t)=0xt-1e-x𝑑x.

  3. 3.

    The expected valueMathworldPlanetmath of a gamma random variable is given by E[X]=αλ, and the varianceMathworldPlanetmath by Var[X]=αλ2

  4. 4.

    The moment generating function of a gamma random variable is given by MX(t)=(λλ-t)α.

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 λ is a gamma (Erlang) variate with parameters n,λ.

Title gamma random variable
Canonical name GammaRandomVariable
Date of creation 2013-03-22 11:54:27
Last modified on 2013-03-22 11:54:27
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Definition
Classification msc 60-00
Classification msc 62-00
Synonym gamma distribution
Defines Erlang random variable