Gamma Distribution

The gamma function is defined by . Note that as , and . Since much faster than , the improper integral converges and is well defined.

Integration by parts yields for each x. This identity yields for each positive integer n.

The random variable X is said to have a gamma distribution with parameters provided

Graph of f

http://www.causeweb.org/repository/statjava/GammaDensityApplet.html

a) When , the gamma distribution is just an exponential distribution with parameter .

b) If the random variable W has a Poisson distribution with parameter and the random variable is the time until exactly n events for the Poisson process have occurred, then has a gamma distribution with parameters .

c) The distribution with n degrees of freedom is an important distribution in statistics and it is just a gamma distribution with parameters .