Page 198 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 181

Answer: let X be the number of eggs laid by the insect, and Y be the number
of eggs developed. Then, given X ˆ r, the distribution of Y is binomial with
parameters r and p. Thus,
r k r k

P Y ˆ kjX ˆ r†ˆ p 1 p† ; k ˆ 0; 1; ... ; r:
k
Now, using the total probability theorem, Theorem 2.1 [Equation (2.27)],
1
X
P Y ˆ k†ˆ P Y ˆ kjX ˆ r†P X ˆ r†
rˆk 6:49†
1
X k r k r
r p 1 p†
e
ˆ :
k r!
rˆk
If we let r ˆ k ‡ n, Equation (6.49) becomes
1 k n n‡k
X n ‡ k p 1 p† e
P Y ˆ k†ˆ
k n ‡ k†!
nˆ0
n n
p † e 1 1 p†
k X
ˆ
k! n!
nˆ0
k p
k 1 p†
p † e e p † e
ˆ ˆ ; k ˆ 0; 1; 2; ... :
k! k!
6:50†
An important observation can be made based on this result. It implies that, if

a random variable X is Poisson distributed with parameter , then a random
variable Y , which is derived from X by selecting only with probability p each

of the items counted by X, is also Poisson distributed with parameter p . Other
examples of the application of this result include situations in which Y is the
number of disaster-level hurricanes when X is the total number of hurricanes
occurring in a given year, or Y is the number of passengers not being able to board
a given flight, owing to overbooking, when X is the number of passenger arrivals.

6.3.1 SPATIAL DISTRIBUTIONS

The Poisson distribution has been derived based on arrivals developing in time,
but the same argument applies to distribution of points in space. Consider the
distribution of flaws in a material. The number of flaws in a given volume has
a Poisson distribution if Assumptions 1–3 are valid, with time intervals replaced by

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