Introduction to Probability and Statistics for Engineers and Scientists

112 Chapter 4:Random Variables and Expectation

value ofX, but the expected value of some function ofX, sayg(X). How do we go
about doing this? One way is as follows. Sinceg(X) is itself a random variable, it must
have a probability distribution, which should be computable from a knowledge of the
distribution ofX. Once we have obtained the distribution ofg(X), we can then compute
E[g(X)]by the definition of the expectation.

EXAMPLE 4.5a SupposeXhas the following probability mass function

p(0)=.2, p(1)=.5, p(2)=.3

CalculateE[X^2 ].

SOLUTION LettingY =X^2 , we have thatYis a random variable that can take on one of
the values 0^2 ,1^2 ,2^2 with respective probabilities

pY(0)=P{Y= 02 }=.2

pY(1)=P{Y= 12 }=.5

pY(4)=P{Y= 22 }=.3

Hence,

E[X^2 ]=E[Y]=0(.2)+1(.5)+4(.3)=1.7 ■

EXAMPLE 4.5b The time, in hours, it takes to locate and repair an electrical breakdown in
a certain factory is a random variable — call itX— whose density function is given by

fX(x)=

{

1if0<x< 1

0 otherwise

If the cost involved in a breakdown of durationxisx^3 , what is the expected cost of such
a breakdown?

SOLUTION LettingY =X^3 denote the cost, we first calculate its distribution function as
follows. For 0≤a≤1,

FY(a)=P{Y≤a}

=P{X^3 ≤a}

=P{X≤a1/3}

=

∫a1/3

0

dx

=a1/3