Engineering Optimization: Theory and Practice, Fourth Edition

634 Stochastic Programming

the range−∞to∞. Such a quantity (likeX)is called arandom variable. We denote

a random variable by a capital letter and the particular value taken by it by a lowercase

letter. Random variables are of two types: (1) discrete and (2) continuous. If the random

variable is allowed to take only discrete valuesx 1 , x 2 ,... , xn, it is called adiscrete

random variable. On the other hand, if the random variable is permitted to take any

real value in a specified range, it is called acontinuousrandom variable. For example,

the number of vehicles crossing a bridge in a day is a discrete random variable,

whereas the yield strength of steel can be treated as a continuous random variable.

Probability Mass Function (for Discrete Random Variables). Corresponding to each

xithat a discrete random variableXcan take, we can associate a probability of occur-

renceP (xi) We can describe the probabilities associated with the random variable. X

by a table of values, but it will be easier to write a general formula that permits one to

calculateP (xi) y substituting the appropriate value ofb xi. Such a formula is called the

probability mass functionof the random variableXand is usually denoted asfX(xi),

or simply asf (xi) Thus the function that gives the probability of realizing the random.

variableX=xiis called the probability mass functionfX(xi) Therefore,.

f (xi)=fX(xi) =P(X=xi) (11.4)

Cumulative Distribution Function (Discrete Case). Although a random variableX

is described completely by the probability mass function, it is often convenient to

deal with another, related function known as theprobability distribution function. The

probability that the value of the random variableXis less than or equal to some number

xis defined as thecumulative distribution functionFX(x).

FX(x) =P(X≤x)=

∑

i

fX(xi) (11.5)

where summation expends over those values ofisuch thatxi≤ x.Since the distribu-

tion function is a cumulative probability, it is also called the cumulative distribution

function.

Example 11.1 Find the probability mass and distribution functions for the number

realized when a fair die is thrown.

SOLUTION Since each face is equally likely to show up, the probability of realizing

any number between 1 and 6 is^16.

P(X= 1 )=P (X= 2 )= · · · =P (X= 6 )=^16

fX( 1 )=fX( 2 )=· · · =fX( 6 )=^16

The analytical form ofFX( )x is

FX(x)=

x

6

for 1≤x≤ 6

It can be seen that for any discrete random variable, the distribution function will

be a step function. If the least possible value of a variableXisSand the greatest