Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.8. Expectation of a Random Variable 61

Definition 1.8.1(Expectation).LetXbe a random variable. IfXis a continuous
random variable with pdff(x)and
∫∞

−∞

|x|f(x)dx <∞,

then theexpectationofXis

E(X)=

∫∞

−∞

xf(x)dx.

IfXis a discrete random variable with pmfp(x)and
∑

x

|x|p(x)<∞,

then theexpectationofXis

E(X)=

∑

x

xp(x).

Sometimes the expectationE(X) is called themathematical expectationof
X,theexpected valueofX,orthemeanofX. When the mean designation is
used, we often denote theE(X)byμ; i.e,μ=E(X).

Example 1.8.1(Expectation of a Constant).Consider a constant random variable,
that is, a random variable with all its mass at a constantk. This is a discrete random
variable with pmfp(k) = 1. We have by definition that

E(k)=kp(k)=k. (1.8.1)

Example 1.8.2.Let the random variableXof the discrete type have the pmf given

by the table

x 1234

p(x) 104 101 103 102

Herep(x)=0ifxis not equal to one of the first four positive integers. This

illustrates the fact that there is no need to have a formula to describe a pmf. We

have

E(X)=(1)

(

4

10

)

+(2)

(

1

10

)

+(3)

(

3

10

)

+(4)

(

2

10

)

=

23

10

=2. 3.

Example 1.8.3.Let the continuous random variableXhave the pdf

f(x)=

{

4 x^30 <x< 1

0elsewhere.

Then

E(X)=

∫ 1

0

x(4x^3 )dx=

∫ 1

0

4 x^4 dx=

4 x^5

5

∣

∣

∣

∣

1

0

=

4

5

.