Engineering Optimization: Theory and Practice, Fourth Edition

11.2 Basic Concepts of Probability Theory 637

Figure 11.2 Two density functions with same mean.

SOLUTION

X=

∑^6

i= 0

xipX(xi) = 0 ( 0. 02 )+ 1 ( 0. 15 )+ 2 ( 0. 22 )+ 3 ( 0. 26 )

+ 4 ( 0. 17 )+ 5 ( 0. 14 )+ 6 ( 0. 04 )

= 2. 99

X^2 =

∑^6

i= 0

x^2 ipX(xi) = 0 ( 0. 02 )+ 1 ( 0. 15 )+ 4 ( 0. 22 )+ 9 ( 0. 26 )

+ 16 ( 0. 17 )+ 25 ( 0. 14 )+ 36 ( 0. 04 )

= 11. 03

Thus
σX^2 =X^2 −(X)^2 = 11. 03 −( 2. 99 )^2 = 2. 0 899 or σX= 1. 4456

Example 11.3 The force applied on an engine brake (X)is given by

fX(x)=











x

48

, 0 ≤x≤8 lb

12 −x

24

, 8 ≤x≤12 lb

Determine the mean and standard deviation of the force applied on the brake.

SOLUTION

μX=E[X]=

∫∞

−∞

xfX(x) dx=

∫ 8

0

x

x

48

dx+

∫ 12

8

x

12 −x

24

dx= 6. 6667

E[X^2 ]=

∫∞

−∞

x^2 fX(x) dx=

∫ 8

0

x^2

x

48

dx+

∫ 12

8

x^2

12 −x

24

dx