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1 Specification of the gamma distribution

1.1 Probability density function

The probability density function of the gamma distribution 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.

Alternatively, the gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ:

1.2 Cumulative distribution function

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

2 Properties

2.1 Moments

Let X be a random variable following a gamma distribution with parameters k and θ. Then X has the following properties:

mode

for k ≥ 1

mean

variance

skewness

kurtosis

2.2 Relation to other distributions

If X₁ has a gamma distribution with parameters k₁ and θ, and X₂ has a gamma distribution with parameters k₂ and θ, then X₁ + X₂ has a gamma distribution with parameters k₁ + k₂ and θ.

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 distributionThe Erlang distribution is a probability distribution developed by A. Erlang to predict waiting times in queuing systems, particularly in the case of telephone traffic engineering. The Erlang distribution is the distribution of the sum of independent iden (so named in honor of A. K. ErlangAgner Krarup Erlang ( January 1, 1878 February 3, 1929) was a Danish mathematician, statistician, and engineer who invented the fields of queueing theory and traffic engineering. Erlang was born at Lonborg (Lonborg), near Tarm, in Jutland. He was the son) and is the probability distribution of the waiting time of the kth "arrival" in a one-dimensional Poisson processA Poisson process one of a variety of things named after the French mathematician Simeon-Denis Poisson (1781 1840), is a stochastic process that assigns to each bounded interval of time or to each bounded region in some space (for example, a Euclidean pla with intensity 1/θ.

If k is a half-integer and θ = 2, then the gamma distribution is a chi-square distributionFor any positive integer , the chi-square distribution with k degrees of freedom is the probability distribution of the random variable : where Z . Z are independent normal variables, each having expected value 0 and variance 1. This distribution is usual with 2 k degrees of freedom.

The gamma distributions are infinitely divisibleThe concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of probability distributions.