Engineering Optimization: Theory and Practice, Fourth Edition

638 Stochastic Programming

= 21. 3333 + 29. 3333 = 50. 6666

σX^2 =E[X^2 ]−(E[X])^2 = 05. 6666 −( 6. 6667 )^2

= 6. 2 222 or σX= 2. 4944

11.2.4 Function of a Random Variable

IfXis a random variable, any other variableYdefined as a function ofXwill also be

a random variable. IffX(x) andFX(x) denote, respectively, the probability density and

distribution functions ofX, the problem is to find the density functionfY(y) and the

distribution functionFY(y) of the random variableY. Let the functional relation be

Y=g(X) (11.15)

By definition, the distribution function ofYis the probability of realizingYless than

or equal toy:

FY(y) =P(Y≤y)=P (g≤y)

=

∫

g(x)≤y

fX(x) dx (11.16)

where the integration is to be done over all values ofxfor whichg(x)≤y.

For example, if the functional relation betweenyandxis as shown in Fig. 11.3,

the range of integration is shown asx 1 + x 2 + x 3 + · · · .The probability density

function ofYis given by

fY(y)=

∂

∂y

[FY(y)] (11.17)

IfY=g(X), the mean and variance ofYare defined, respectively, by

E(Y )=

∫∞

−∞

g(x)fX(x) dx (11.18)

Var[Y]=

∫∞

−∞

[ (x)g −E(Y )]^2 fX(x) dx (11.19)

Figure 11.3 Range of integration in Eq. (11.16).