Gamma Distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. There are three different parameterizations in common use:

1. With a shape parameter k and a scale parameter θ.
2. With a shape parameter α = k and an inverse scale parameter β = 1⁄_θ, called a rate parameter.
3. With a mean parameter and the shape parameter k.

The parameterization with k and θ appears to be more common in econometrics and certain other applied fields, where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution.

The parameterization with α and β is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution — or for that matter, the β of the gamma distribution itself. (The closely related inverse gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.)

If k is an integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially distributed random variables, each of which has a mean of θ (which is equivalent to a rate parameter of 1/θ).

The gamma distribution is the maximum entropy probability distribution for a random variable X for which is fixed and greater than zero, and is fixed ( is the digamma function).