Page 197 - Fundamentals of Probability and Statistics for Engineers

180 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. For example, function POISSON in
1 TM
Microsoft Excel 2000 gives the Poisson probabilities given by Equation
(6.44) (see Appendix B).

Example 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 Y ˆ X 1 ‡ X 2 . Determine the distribution of Y .
Answer: we proceed by determining first the characteristic functions of X 1
and X 2 . They are
1 jtk k
X e
t†ˆ Efe jtX 1 gˆ e 1 1
X 1
k!
kˆ0
jt
ˆ exp‰ 1 e 1†Š
and

jt
t†ˆ exp‰ 2 e 1†Š:
X 2
Owing to independence, the characteristic function of Y, Y (t), is simply the
(t) (t) [see Equation (4.71)]. Hence,
product of X 1 and X 2
jt
t†ˆ exp‰ 1 ‡ 2 † e 1†Š:
Y t†ˆ X 1 t† X 2
By inspection, it is the characteristic function corresponding to a Poisson
distribution with parameter 1 ‡ 2 . Its pmf is thus

k
1 ‡ 2 † exp‰ 1 ‡ 2 †Š
p k†ˆ ; k ˆ 0; 1; 2; ... : 6:48†
Y
k!
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.
Example 6.13. Problem: suppose that the probability of an insect laying r

r
eggs is e /r!, r ˆ 0, 1, .. ., and that the probability of an egg developing is p.
Assuming mutual independence of individual developing processes, show that
k p
the probability of a total of k survivors is (p ) e /k!.

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