Fundamentals of Probability and Statistics for Engineers

it be given that, on average, a telephone user is on the phone 5 minutes per
hour; an estimate of p is

The solution to this problem is given by

Ex ample 6. 3. Problem: let X 1 and X 2 be two independent random variables,
both having binomial distributions with parameters (n 1 ,p)and (n 2 , p), respect-
ively, and let Determine the distribution of random variable Y.
Answer: the characteristic functions of X 1 and X 2 are, according to the first
of Equations (6.8),

In view of Equation (4.71), the characteristic function of Y is simply the
product of and Thus,

By inspection, it is the characteristic function corresponding to a binomial
distribution with parameters H ence, we have

Generalizing the answer to Example 6.3, we have the following important
result as stated in Theorem 6.1.

Theorem 6.1:The binomial distribution generates itself under addition of
independent random variables with the same p.

Ex ample 6. 4. Problem: if random variables X and Y are independent binomial
distributed random variables with parameters (n 1 ,p)and (n 2 ,p), determine the
conditional probability mass function of X given that

166 Fundamentals of Probability and Statistics for Engineers

pˆ

5

60

ˆ

1

12

:

pX… 2 †‡pX… 3 †ˆ

3

2



1

12

 2

11

12



‡

3

3



1

12

 3

ˆ

11

864

ˆ 0 : 0197 :

YˆX 1 ‡X 2.

X 1 …t†ˆ…pejt‡q†n^1 ;X 2 …t†ˆ…pejt‡q†n^2 :

X 1 (t) X 2 (t).

Y…t†ˆX 1 …t†X 2 …t†

ˆ…pejt‡q†n^1 ‡n^2 :

(n 1 ‡n 2 ,p).

pY…k†ˆ

n 1 ‡n 2

k



pkqn^1 ‡n^2 k; kˆ 0 ; 1 ;...;n 1 ‡n 2 :

X‡Yˆm; 0 mn 1 ‡n 2 :