Interactive animation

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

X = i Xi

is  G(a, l) with :

a = Si ai

Sufficient statistic

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

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Tutorial

 

 These results are demonstrated in the following Tutorial.

 

BASIC PROPERTIES OF THE GAMMA DISTRIBUTION

Moment generating function

General case

Special case : a is an integer

Mean µ

Mode

Variance s²

Additivity

TUTORIAL

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* One adjustable exponential distribution.
* Adjustable sample size.
* Progressive histogramm of the sum of the observations.

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Related readings :

The exponential distribution

The Chi-square distribution

Download this Glossary

 

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