Page 99 - Fundamentals of Probability and Statistics for Engineers

82 Fundamentals of Probability and Statistics for Engineers

Now,
n
n X X
n
2
2
X
EfY gˆ k p k†ˆ k k 1†p k†‡ kp k†;
Y Y Y
kˆ0 kˆ0 kˆ0
and
n
X
kp k†ˆ nq:
Y
kˆ0
Proceeding as in Example (4.1),
n n
X 2 X n 2†! k 2 n k
k k 1†p k†ˆ n n 1†q q 1 q†
Y k 2†! n k†!
kˆ0 kˆ2
X m !
m
j
ˆ n n 1†q 2 q 1 q† m j
j
jˆ0
2
ˆ n n 1†q :
Thus,
2
2
EfY gˆ n n 1†q ‡ nq;
and
2
2
2
ˆ n n 1†q ‡ nq nq† ˆ nq 1 q†:
Y
Example 4.6. We again use Equation (4.8) to determine the variance of X

defined in Example 4.2. The second moment of X is, on integrating by parts,
Z 1 1
2 2x
2
EfX gˆ 2 x e dx ˆ :
0 2
Hence,
1 1 1
2
2
2
ˆ EfX g m ˆ ˆ :
X X
2 4 4
Example 4.7. Problem: owing to inherent manufacturing and scaling inaccura-
cies, the tape measures manufactured by a certain company have a standard
deviation of 0.03 feet for a three-foot tape measure. What is a reasonable
estimate of the standard deviation associated with three-yard tape measures
made by the same manufacturer?

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