Introduction to Probability and Statistics for Engineers and Scientists

4.4Expectation 107

EXAMPLE 4.3h The joint density ofXandYis given by

f(x,y)=

{ 12

5 x(2−x−y)0<x<1, 0<y<^1

0 otherwise

Compute the conditional density ofX, given thatY=y, where 0<y<1.

SOLUTION For 0<x<1, 0<y<1, we have

fX|Y(x|y)=

f(x,y)

fY(y)

=

f(x,y)

∫∞

−∞f(x,y)dx

=

x(2−x−y)

∫ 1

0 x(2−x−y)dx

=

x(2−x−y)

2

3 −y/2

=

6 x(2−x−y)

4 − 3 y

■

4.4Expectation

One of the most important concepts in probability theory is that of the expectation
of a random variable. IfX is a discrete random variable taking on the possible values
x 1 ,x 2 ,..., then theexpectationorexpected valueofX, denoted byE[X], is defined by

E[X]=

∑

i

xiP{X=xi}

In words, the expected value ofXis a weighted average of the possible values thatXcan
take on, each value being weighted by the probability thatXassumes it. For instance, if
the probability mass function ofXis given by

p(0)=^12 =p(1)

then

E[X]= 0

( 1

2

)

+ 1

( 1

2

)

=^12

is just the ordinary average of the two possible values 0 and 1 thatXcan assume. On the
other hand, if

p(0)=^13 , p(1)=^23