Fundamentals of Probability and Statistics for Engineers

Now,

and

Proceeding as in Example (4.1),

Thus,

and

Ex ample 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,

H ence,

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?

82 Fundamentals of Probability and Statistics for Engineers

EfY^2 gˆ

Xn

kˆ 0

k^2 pY…k†ˆ

Xn

kˆ 0

k…k 1 †pY…k†‡

Xn

kˆ 0

kpY…k†;

Xn

kˆ 0

kpY…k†ˆnq:

Xn

kˆ 0

k…k 1 †pY…k†ˆn…n 1 †q^2

Xn

kˆ 2

…n 2 †!

…k 2 †!…nk†!

qk^2 … 1 q†nk

ˆn…n 1 †q^2

Xm

jˆ 0

m

j

!

qj… 1 q†mj

ˆn…n 1 †q^2 :

EfY^2 gˆn…n 1 †q^2 ‡nq;

^2 Yˆn…n 1 †q^2 ‡nq…nq†^2 ˆnq… 1 q†:

EfX^2 gˆ 2

Z 1

0

x^2 e^2 xdxˆ

1

2

:

^2 XˆEfX^2 gm^2 Xˆ

1

2



1

4

ˆ

1

4

: