GAMMA

IMPORTANT CONTINUOUS RANDOM VARIABLE [brown line]

Used to model time required until occurance of event A.

Has probability density function (pdf)

The Gamma RV has different shapes based on the values of alpha and beta.

Click on a button to see an example graph at right.
Gamma pdf pict.


In general, the cumulative distribution function (CDF) of an Gamma random variable does not have a closed form expression. For certain special cases of alpha and beta there are closed form solutions.


[brown line]

CASES OF ALPHA

If see Exponential RV.

If then the pdf is and can be illustrated as

Gamma CDF pict

[brown line]

PROPERTIES

Mean:


Variance:


rth central moment about the origin:


Moment Generating Function:





  • If and then we get an Exponential Random Variable.

  • If and then we get a Chi-square Random Variable.

  • If is an integer, m, then we get a m-Erlang Random Variable.

  • If is an integer, the Gamma CDF is related to the Poisson CDF.
[brown line]

RELATED
CONCEPTS

  Random Variables
Probability Density Function
Cumulative Distribution Function
Important Continuous Random Variables
[brown line]

Copyright © 1998 Rensselaer Polytechnic Institute. All Rights Reserved.