GAMMA/ERLANG AND POISSON

Analysis of the relationship between the two random variables [brown line]

A Gamma or Erlang Cumulative Distribution Function (CDF) is related to the Poisson CDF.

Here is the derivation:

[brown line]

A Gamma RV has the probability density function (pdf) If alpha is an integer, then
This makes the Random Variable an Erlang RV with the pdf:

The integral (Eq1) below be verified by integration by parts r times to be equal to the sum This equals

if Y is a Poisson Random Variable with parameter a.

Then recall the Gamma (Erlang) CDF which can be written as Letting u = y/beta, the CDF becomes Substituting the integral from Eq1 above results in Therefore, the CDF of the Gamma distribution may be written in terms of the tabulated CDF of the Poisson distribution.

[brown line]

RELATED
CONCEPTS

Gamma Random Variables
Erlang Random Variables
Poisson Random Variables
Cumulative Distribution Function
[brown line]

Copyright © 1998 Rensselaer Polytechnic Institute. All Rights Reserved.