Engineering Optimization: Theory and Practice, Fourth Edition

640 Stochastic Programming

exclusive ways of obtaining the points lying betweenxandx+dx. Let the lower and

upper limits of y bea 1 (x) andb 1 (x) Then.

P[x≤x′≤ x+dx]=

[∫b 1 (x)

a 1 (x)

fX,Y(x, y) dy

]

dx=fX(x) dx

fX(x)=

∫y 2 =b 1 (x)

y 1 =a 1 (x)

fX,Y(x, y) dy (11.25)

Similarly, we can show that

fY(y)=

∫x 2 =b 2 (y)

x 1 =a 2 (y)

fX,Y(x, y) dx (11.26)

11.2.6 Covariance and Correlation

IfXandYare two jointly distributed random variables, the variances ofXandYare

defined as

E[(X−X)^2 ] =Var[X]=

∫∞

−∞

(x−X)^2 fX(x) dx (11.27)

E[(Y−Y )^2 ] =Var[Y]=

∫∞

−∞

(y−Y )^2 fY(y) dy (11.28)

and the covariance ofXandYas

E[(X−X)(Y−Y )]=Cov(X, Y )

=

∫∞

−∞

∫∞

−∞

(x−X)(y−Y )fX,Y(x, y)dxdy

=σX,Y (11.29)

The correlation coefficient,ρX,Y, for the random variables is defined as

ρX,Y=

Cov(X, Y )

σXσY

(11.30)

and it can be proved that− 1 ≤ρX,Y≤. 1

11.2.7 Functions of Several Random Variables

IfYis a function of several random variablesX 1 , X 2 ,... , Xn, the distribution and den-

sity functions ofYcan be found in terms of the joint density function ofX 1 , X 2 ,... , Xn

as follows:

Let

Y=g(X 1 , X 2 ,... , Xn) (11.31)