Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 329

From ( 5.159 ) and ( 5.160 ), the variance is

X 2

s 2

2 2

¼ X 2 ¼ðltÞ þ lt ðltÞ ¼ lt ¼ k: (5.161)

This result shows an interesting property of a Poisson variable:
The variance of a Poisson random variable is equal to its mean value .
Next section shows another interesting property of Poisson random variable.

5.7.5 Sum of Independent Poisson Variables

Co nsid er two independent Poisson random variables X 1 and X 2 with mean values of
k 1 and k 2 , respectively:

X ¼ X 1 þ X 2 : (5.162)

The characteristic function of the sum ( 5.162 ) is equal to the product of the
corresponding characteristic functions,

f X ðoÞ¼f X 1 ðoÞf X 2 ðoÞ: (5.163)

Using the expression for the characteristic function ( 5.158 ), we have:

f X ðoÞ¼ e k 1 ð e jo

1 Þ e k 2 ð e jo 1 Þ ¼ e ðk 1 þ k 2 Þð e jo 1 Þ ¼ e kð e jo

1 Þ :
(5.164)

Comparing ( 5.158 ) and ( 5.164 ), we can conclude that the variable X is also a
Poisson random variable and that the following is true:

The sum of independent Poisson variables is a Poisson variable .

The parameter of the X variable is,

k ¼ k 1 þ k 2 ; (5.165)

where k 1 and k 2 are parameters (mean values) of the variables X 1 and X 2 ,
respectively.