Engineering Optimization: Theory and Practice, Fourth Edition

636 Stochastic Programming

Discrete Case. Let us assume that there arentrials in which the random variable

Xis observed to take on the valuex 1 (n 1 times),x 2 (n 2 times), and so on, and

n 1 +n 2 + · · · +nm= n.Then the arithmetic mean ofX, denoted asX, is given by

X=

∑m

k= 1 xknk

n

=

∑m

k= 1

xk

nk

n

=

∑m

k= 1

xkfX(xk) (11.9)

wherenk/n s the relative frequency of occurrence ofi xkand is same as the probability

mass functionfX(xk) Hence in general, the expected value,. E(X), of a discrete random

variable can be expressed as

X=E(X)=

∑

i

xifX(xi), sum over alli (11.10)

Continuous Case. IffX(x) s the density function of a continuous random variable,i

X, the mean is given by

X=μx= E(X)=

∫∞

−∞

xfX(x) dx (11.11)

Standard Deviation. The expected value or mean is a measure of the central

tendency, indicating the location of a distribution on some coordinate axis. A measure

of the variability of the random variable is usually given by a quantity known as the

standard deviation. The mean-square deviation or variance of a random variableXis

defined as

σX^2 = arV (X)=E[(X−μX)^2 ]

=E[X^2 − 2 XμX+μ^2 X]

=E(X^2 )− 2 μXE(X) +E(μ^2 X)

=E(X^2 )−μ^2 X (11.12)

and the standard deviation as

σX= +

√

Var(X)=

√

E(X^2 )−μ^2 X (11.13)

The coefficient of variation (a measure of dispersion in nondimensional form) is

defined as

coefficient of variation ofX=γX=

standard deviation

mean

=

σX

μX

(11.14)

Figure 11.2 shows two density functions with the same meanμXbut with different

variances. As can be seen, the variance measures thebreadthof a density function.

Example 11.2 The number of airplane landings at an airport in a minute (X)and

their probabilities are given by

xi 0 1 2 3 4 5 6

pX(xi) 0.02 0.15 0.22 0.26 0.17 0.14 0.04

Find the mean and standard deviation ofX.