Page 94 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 77

Example 4.1. Problem: From Example 3.9 (page 64), determine the average
number of cars turning west in a group of n cars.
Answer: we wish to determine the mean of Y , EfYg, for which the mass
function is [from Equation (3.51)]

n k n k
p k†ˆ q 1 q† ; k ˆ 0; 1; 2; ... ; n:
Y k
Equation (4.4) then gives
!
n n n
X X k n k
EfYgˆ kp k†ˆ k q 1 q†
Y
kˆ0 kˆ0 k
n
X n! k n k
ˆ q 1 q† :
k 1†! n k†!
kˆ1
Let k 1 ˆ m.Wehave
n 1
X n 1 m n 1 m
EfYgˆ nq q 1 q† :
m
mˆ0
The sum in this expressions is simply the sum of binomial probabilities and
hence equals one. Therefore,

EfYgˆ nq;

which has a numerical value since n and q are known constants.
Example 4.2. Problem: the waiting time X (in minutes) of a customer waiting

to be served at a ticket counter has the density function
2x
2e ; for x 0;
f x†ˆ
X
0; elsewhere:
Determine the average waiting time.
Answer: referring to Equation (4.5), we have, using integration by parts,
Z 1 1
EfXgˆ x 2e 2x †dx ˆ minute:
0 2

Example 4.3. Problem: from Example 3.10 (pages 65), find the average
resistance of the resistors after screening.

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