Fundamentals of Probability and Statistics for Engineers

the PDF associated with a Poisson–distributed random variable. The answer
to Example 6.11, for example, can easily be read off from Figure 6.4. We
mention again that a large number of computer software packages are
available to produce these probabilities. F or example, function POISSON in
Microsoft ExcelTM2000 gives the Poissonprobabilities given by Equation
(6.44) (see Appendix B).

Ex ample 6. 12. Problem: let X 1 and X 2 be two independent random variables,
both having Poisson distributions with parameters 1 and 2 , respectively, and
let Determine the distribution of Y.
Answer: we proceed by de te rmining first the characteristic functions of X 1
and X 2. They are

and

Owing to independence, the characteristic function of is simply the
product of and [see Equation (4.71)]. H ence,

By inspection, it is the characteristic function corresponding to a Poisson
distribution with parameter Its pmf is thus

As in the case of the binomial distribution, this result leads to the following
important theorem, Theorem 6.2.

Theorem 6.2:the Poisson distribution generates itself under addition of
independent random variables.

Ex ample 6. 13. Problem: suppose that the probability of an in sect laying r
eggs is 0,1,..., and that the probability of an egg de veloping is p.
Assuming mutual independence of individual developing processes, show that
the probability of a total of k survivors is

180 Fundamentals of Probability and Statistics for Engineers

1



YˆX 1 ‡X 2.

X 1 …t†ˆEfejtX^1 gˆe^1

X^1

kˆ 0

ejtk 1 k

k!

ˆexp‰ 1 …ejt 1 †Š

X 2 …t†ˆexp‰ 2 …ejt 1 †Š:

Y,Y(t),

X 1 (t) X 2 (t)

Y…t†ˆX 1 …t†X 2 …t†ˆexp‰… 1 ‡ 2 †…ejt 1 †Š:

 1 ‡ 2.

pY…k†ˆ

… 1 ‡ 2 †kexp‰… 1 ‡ 2 †Š

k!

; kˆ 0 ; 1 ; 2 ;...: … 6 : 48 †

re/r!,rˆ

(p)kep/k!.