A.4 Gamma distribution

X has a (one-parameter) gamma distribution with parameter α, denoted

if it has density

This is simply another name for the distribution, we refer to as

If , then Y has a two-parameter gamma distribution with parameters α and β denoted

and it has density

so that its mean and variance are

This is simply another name for the distribution, we refer to as

If we recover the one-parameter gamma distribution; if so that the density is

we obtain another special case sometimes called the (negative) exponential distribution and denoted

The distribution function of any variable with a gamma distribution is easily found in terms of the incomplete gamma function

or in terms of Karl Pearson’s incomplete gamma function

Extensive tables can be found in Pearson (1924).