Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 165The moments and distribution of X can be easily found by using Equation(6.10). Sinceit follows from Equation (4.38) thatEfX j gˆ0…q†‡1…p† ˆp; j ˆ 1; 2; ...; n;EfXg ˆp ‡ 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 followingour discussion in Section 4.4 concerning sums of independent randomvariables.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 arefurther illustrated by the following additional examples.Example 6.1. Problem: a homeowner has just installed 20 light bulbs in a newhome. Suppose that each has a probability 0.2 of functioning more than threemonths. What is the probability that at least five of these function more thanthree months? What is the average number of bulbs the homeowner has toreplace in three months?Answer: it is reasonable to assume that the light bulbs perform independently.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 answerto the first question is thus given byX 20kˆ5p X …k† ˆ1ˆ 1ˆ 1X 4kˆ0X 4kˆ0p X …k† 20…0:2† k …0:8† 20kk…0:012 ‡ 0:058 ‡ 0:137 ‡ 0:205 ‡ 0:218† ˆ0:37:The average number of replacements is20 EfXg ˆ20 np ˆ 20 20…0:2† ˆ16:Example 6.2. Suppose that three telephone users use the same number andthat we are interested in estimating the probability that more than one will useit at the same time. If independence of telephone habit is assumed, the probabilityof exactly k persons requiring use of the telephone at the same time isgiven by the mass function p X (k) associated with the binomial distribution. LetTLFeBOOK