Engineering Optimization: Theory and Practice, Fourth Edition

642 Stochastic Programming

These results can be generalized to the case whenYis a linear function of several

random variables. Thus if

Y=

∑n

i= 1

aiXi (11.38)

then

E(Y )=

∑n

i= 1

aiE(Xi) (11.39)

Var(Y )=

∑n

i= 1

ai^2 Var (Xi)+

∑n

i= 1

∑n

j= 1

aiajCov (Xi, Xj), i=j (11.40)

Approximate Mean and Variance of a Function of Several Random Variables.

IfY=g(X 1 ,... , Xn) the approximate mean and variance of, Ycan be obtained as

follows. Expand the functiongin a Taylor series about the mean valuesX 1 ,X 2 ,... ,Xn

to obtain

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

∑n

i= 1

(Xi−Xi)

∂g

∂Xi

+

1

2

∑n

i= 1

∑n

j= 1

(Xi−Xi) (Xj−Xj)

∂^2 g

∂Xi∂Xj

+ · · · (11.41)

where the derivatives are evaluated at(X 1 ,X 2 ,... ,Xn) By truncating the series at.

the linear terms, we obtain the first-order approximation toYas

Y≃g(X 1 ,X 2 ,... ,Xn)+

∑n

i= 1

(Xi−Xi)

∂g

∂Xi

∣

∣

∣

∣

(X 1 ,X 2 ,...,Xn)

(11.42)

The mean and variance ofYgiven by Eq. (11.42) can now be expressed as [using

Eqs. (11.39) and (11.40)]

E(Y )≃g(X 1 ,X 2 ,... ,Xn) (11.43)

Var(Y )≃

∑n

i= 1

c^2 iVar (Xi)+

∑n

i= 1

∑n

j= 1

cicjCov (Xi, Xj), i=j (11.44)

whereciandcjare the values of the partial derivatives∂g/∂Xiand ∂g/∂Xj, respec-

tively, evaluated at(X 1 ,X 2 ,... ,Xn).

It is worth noting at this stage that the approximation given by Eq. (11.42)

is frequently used in most of the practical problems to simplify the computations

involved.