Page 219 - Fundamentals of Probability and Statistics for Engineers

202 Fundamentals of Probability and Statistics for Engineers

Theorem 7.2: Let X be a normal random variable with distribution N(m, 2 ).

Then (X m is the standardized normal random variable with distribution
)/
N(0, 1), or
X m
U ˆ : 7:22†

Proof of Theorem 7.2: the characteristic function of random variable
"X m)/ is
jt X m† jtm jtX jtm

E exp ˆ exp E exp ˆ exp X t= †:

From Equation (7.12) we have

2 2
t
X t†ˆ exp jmt : 7:23†
2

Hence,

2 2
jt X m† jmt jmt t
E exp ˆ exp ‡
2
2
t
ˆ exp : 7:24†
2
The result given above takes the form of X "t) with m ˆ 0 and ˆ 1, and the
proof is complete.
Theorem 7.2 implies that

a m b m

P a < X b†ˆ P‰a < U ‡ m† bŠˆ P < U : 7:25†

The value of the right-hand side can now be found from Table A.3, with the aid
of Equation (7.21) if necessary.
As has been noted, probabilities provided by Table A.3 can also be obtained
from a number of computer software packages such as Microsoft 1 Excel TM
2000 (see Appendix B).
Example 7.3. Problem: owing to many independent error sources, the length
of a manufactured machine part is normally distributed with m ˆ 11 cm and
ˆ 0 2 cm. If specifications require that the length be between 10.6 cm
.

TLFeBOOK