Gamma distribution

Gamma distribution

In probability theory and statistics, the gamma distribution is a continuous probability distribution. Its probability density function can be expressed in terms of the gamma function:

where k > 0 is the shape parameter and θ > 0 is the scale parameter of the gamma distribution.

The cumulative distribution function can be expressed in terms of the incomplete gamma function,

The expected value and variance of a gamma random variable X are:

If has a gamma distribution with parameters and θ, and has a gamma distribution with parameters and &theta, then has a gamma distribution with parameters and &theta.

If k is equal to 1, the gamma distribution is an exponential distribution with parameter θ. The sum of n exponential variables, all with the same parameter θ, is a gamma variable with parameters n and θ.

If k is an integer, the gamma distribution is an Erlang distribution (so named in honor of A.K. Erlang) and is the probability distribution of the waiting time of the kth "arrival" in a one-dimensional Poisson process with intensity 1/θ.

If k is a half-integer and θ = 2, then the gamma distribution is a chi-square distribution with 2 k degrees of freedom.

References

R.V. Hogg and A.T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)