Gamma  (distribution)

Also known as "Erlang's distribution".

Definition of the Gamma distribution

The Gamma distribution is a continuous distribution whose probability density function p(x) is :
 

    * p(x) = 0   for x < 0,

    * For x 0 :

where G(a ) is the Gamma function.

    * l is any positive real number.

    * a is usually an integer (a = 1, 2, ...).   

It is therefore a family of distributions indexed by the two parameters a and l. We will denote the Gamma distribution G(a, l).

This definition may look rather arbitrary, but in fact the Gamma distribution G(n, l)  turns up naturally as the distribution of the sum of n independent exponential random variables, all with the same parameter l (for a proof, see Tutorial below. Also see interactive animation  ).

 Make n = 1, and the exponential distribution appears as a special case of the Gamma distribution.

Also, the Chi-square distribution with n degrees of freedom is a Gamma distribution with parameters a = n/2 and l = 1/2.

Basic properties of the Gamma distribution

The basic properties of the Gamma distribution are :

Mean

Variance

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This result can be written as :

The significance of this expression will appear when we consider the Gamma distribution as belonging to the natural exponential family. 

Moment generating function

Additivity

If X_i , i = 1, 2, ..., n are independent Gamma random variables with respective parameters (a_i, l), then their sum :

X = _i X_i

is  G(a, l) with :

a = S_i a_i

Sufficient statistic

We identify here a sufficient statistic for the parameter a of the Gamma distribution.

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 These results are demonstrated in the following Tutorial.

BASIC PROPERTIES OF THE GAMMA DISTRIBUTION

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Related readings :