Engineering Optimization: Theory and Practice, Fourth Edition

644 Stochastic Programming

Figure 11.4 Standard normal density function.

By the same token, the values ofzcorresponding top <0.5 can be obtained as

z=φ−^1 (p) =−φ−^1 ( 1 −p) (11.50)

Notice that any normally distributed variable (X)can be reduced to a standard normal

variable by using the transformation

z=

x−μX

σX

(11.51)

For example, ifP (a < X≤b)is required, we have

P (a < X≤b)=

1

σX

√

2 π

∫ b

a

e^ −(^1 /^2 )[(x−μX)/σX]

2

dx (11.52)

By using Eq. (11.51) anddx=σXdz Eq. (11.52) can be rewritten as,

P (a < X≤b)=

1

√

2 π

∫(b−μX)/σX

(a−μX)/σX

e−z

(^2) / 2
dz (11.53)
This integral can be recognized to be the area under the standard normal density curve
between (a−μX)/σXand (b−μX)/σXand hence
P(a < X≤b)=φ

(

b−μX

σX

)

−φ

(

a−μX

σX

)

(11.54)

Example 11.4 The width of a slot on a duralumin forging is normally distributed.

The specification of the slot width is 0. 900 ± 0 .005. The parametersμ=0.9 and

σ= 0 .003 are known from past experience in production process. What is the percent

of scrap forgings?

SOLUTION IfXdenotes the width of the slot on the forging, the usable region is

given by

0. 895 ≤x≤ 0. 905