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

Proof of Theorem 7.2: the characteristic function of random variable

is

F rom Equation (7.12) we have

H ence,

The result given above takes the form of with 0 and 1, and the
proof is complete.
Theorem 7.2 implies that

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 ExcelTM
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 11cm and
0 2 cm. If specifications require that the length be between 10.6 cm

202 Fundamentals of Probability and Statistics for Engineers



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