Applied Statistics and Probability for Engineers

114 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Figure 4-15 Standardizing a normal random variable.

4 7 9 10 13 16 x

–3 –1.5 – 0.5 0 1.5 3 z

11

0.5

0 1.5

Distribution of Z =X σ–μ

Distribution of X

10 13 x

z

Suppose Xis a normal random variable with mean and variance 2. Then,

(4-11)

where Zis a standard normal random variable,and is thez-value

obtained by standardizingX.

The probability is obtained by entering Appendix Table II with z 1 x 2   .

z

1 x 2

P 1 Xx 2 P a

X



x

bP^1 Zz^2

EXAMPLE 4-14 Continuing the previous example, what is the probability that a current measurement is be-

tween 9 and 11 milliamperes? From Fig. 4-15, or by proceeding algebraically, we have

Determine the value for which the probability that a current measurement is below

this value is 0.98. The requested value is shown graphically in Fig. 4-16. We need the value of

xsuch that P(Xx) 0.98. By standardizing, this probability expression can be written as

Appendix Table II is used to find the z-value such that P(Z z) 0.98. The nearest proba-

bility from Table II results in

P 1 Z2.05 2 0.97982

0.98

P 1 Z 1 x 102 22

P 1 Xx 2 P 11 X 102 2  1 x 102 22

0.691460.308540.38292

P 1 0.5Z0.5 2 P 1 Z0.5 2 P 1 Z0.5 2

P 19 X 112 P 119  102 2  1 X 102 2  111  102 22

In the preceding example, the value 13 is transformed to 1.5 by standardizing, and 1.5 is

often referred to as the z-valueassociated with a probability. The following summarizes the

calculation of probabilities derived from normal random variables.

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