Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.8 IMPORTANT DISCRETE DISTRIBUTIONS

1

1

12

2

2 3

3

3 4

4

4 5

5

5 6

6

7

7891011

0

0

0

0

0

0

0. 1 0. 1

0. 1

0. 2

0. 2 0. 2

0. 3

0. 3 0. 3

x

x

x

f(x)

f(x)

f(x)

λ=1 λ=2

λ=5

Figure 30.12 Three Poisson distributions for different values of the parame-

terλ.

The above example illustrates the point that a Poisson distribution typically

rises and then falls. It either has a maximum whenxis equal to the integer part

ofλor, ifλhappens to be an integer, has equal maximal values atx=λ−1and

x=λ. The Poisson distribution always has a long ‘tail’ towards higher values ofX

but the higher the value of the mean the more symmetric the distribution becomes.

Typical Poisson distributions are shown in figure 30.12. Using the definitions of

mean and variance, we may show that, for the Poisson distribution,E[X]=λand

V[X]=λ. Nevertheless, as in the case of the binomial distribution, performing

the relevant summations directly is rather tiresome, and these results are much

more easily proved using the MGF.

The moment generating function for the Poisson distribution

The MGF of the Poisson distribution is given by

MX(t)=E

[

etX

]

=

∑∞

x=0

etxe−λλx

x!

=e−λ

∑∞

x=0

(λet)x

x!

=e−λeλe

t

=eλ(e

t−1)

(30.104)