Page 182 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 165

The moments and distribution of X can be easily found by using Equation
(6.10). Since

EfX j gˆ 0 q†‡ 1 p†ˆ p; j ˆ 1; 2; ... ; n;

it follows from Equation (4.38) that

EfXgˆ p ‡ p ‡ ‡ p ˆ np; 6:11†

which is in agreement with the corresponding expression in Equations (6.8).
Similarly, its variance, characteristic function, and pmf are easily found follow-
ing our discussion in Section 4.4 concerning sums of independent random
variables.
We have seen binomial distributions in Example 3.5 (page 52), Example 3.9
(page 64), and Example 4.11 (page 96). Its applications in other areas are
further illustrated by the following additional examples.
Example 6.1. Problem: a homeowner has just installed 20 light bulbs in a new
home. Suppose that each has a probability 0.2 of functioning more than three
months. What is the probability that at least five of these function more than
three months? What is the average number of bulbs the homeowner has to
replace in three months?
Answer: it is reasonable to assume that the light bulbs perform indepen-
dently. If X is the number of bulbs functioning more than three months
(success), it has a binomial distribution with n ˆ 20 and p ˆ 0:2. The answer
to the first question is thus given by

20 4
X X
p k†ˆ 1 p k†
X X
kˆ5 kˆ0
4
X 20 k 20 k
ˆ 1 0:2† 0:8†
k
kˆ0
ˆ 1 0:012 ‡ 0:058 ‡ 0:137 ‡ 0:205 ‡ 0:218†ˆ 0:37:
The average number of replacements is

20 EfXgˆ 20 np ˆ 20 20 0:2†ˆ 16:

Example 6.2. Suppose that three telephone users use the same number and
that we are interested in estimating the probability that more than one will use
it at the same time. If independence of telephone habit is assumed, the prob-
ability of exactly k persons requiring use of the telephone at the same time is
given by the mass function p (k) associated with the binomial distribution. Let
X

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