Engineering Optimization: Theory and Practice, Fourth Edition

646 Stochastic Programming

=[1−P (Z≤ 1. 667 )]+[1−P (Z≤ 1. 667 )]

= 2. 0 − 2 P (Z≤ 1. 667 )

= 2. 0 − 2 ( 0. 9525 )= 0. 095

= 9 .5 %

Joint Normal Density Function. IfX 1 , X 2 ,... , Xnfollow normal distribution, any

linear function,Y=a 1 X 1 +a 2 X 2 + · · · +anXn, also follows normal distribution with

mean

Y=a 1 X 1 +a 2 X 2 + · · · +anXn (11.55)

and variance

Var(Y )=a 12 Var (X 1 )+a^22 Var (X 2 ) +· · · +an^2 Var (Xn) (11.56)

ifX 1 , X 2 ,... , Xnare independent. In general, the joint normal density function for

n-independent random variables is given by

fX 1 ,X 2 ,...,Xn(x 1 , x 2 ,... , xn)=

1

√

( 2 π )nσ 1 σ 2 · · ·σn

exp



−

1

2

∑n

k= 1

(

xk−Xk

σk

) 2 



=fX 1 (x 1 )fX 2 (x 2 ) · ·· fXn(xn) (11.57)

whereσi=σXi. If the correlation between the random variablesXkandXjis not zero,

the joint density function is given by

fX 1 ,X 2 ,...,Xn(x 1 , x 2 ,... , xn)

=

1

√

( 2 π )n|K|

exp



−^1

2

∑n

j= 1

∑n

k= 1

{K−^1 }j k(xj−Xj)(xk−Xk)



 (11.58)

where

KXjXk=Kj k= E[(xj−Xj)(xk−Xk)]

=

∫∞

−∞

∫∞

−∞

(xj−Xj)(xk−Xk)fXj,Xk(xj, xk)dxjdxk

= onvariance betweenc XjandXk

K =correlation matrix=

     

K 11 K 12 · · · K 1 n

K 21 K 22 · · · K 2 n

..

.

Kn 1 Kn 2 · · · Knn

     

(11.59)

and{K−^1 }j k= jkth element ofK−^1. It is to be noted thatKXjXk= 0 forj=kand =

σX^2 jforj=kin case there is no correlation betweenXjandXk.