Introduction to Probability and Statistics for Engineers and Scientists

124 Chapter 4:Random Variables and Expectation

and so for independentX 1 ,...,Xn,

Var

( n

∑

i= 1

Xi

)

=

∑n

i= 1

Var(Xi)

Proof

We need to prove thatE[XY]=E[X]E[Y]. Now, in the discrete case,

E[XY]=

∑

j

∑

i

xiyjP{X=xi,Y=yj}

=

∑

j

∑

i

xiyjP{X=xi}P{Y=yj} by independence

=

∑

y

yjP{Y=yj}

∑

i

xiP{X=xi}

=E[Y]E[X]

Because a similar argument holds in all other cases, the result is proven.

EXAMPLE 4.7a Compute the variance of the sum obtained when 10 independent rolls of
a fair die are made.

SOLUTION LettingXidenote the outcome of theith roll, we have that

Var

( 10

∑

1

Xi

)

=

∑^10

1

Var(Xi)

= 103512 from Example 4.6a

=^1756 ■

EXAMPLE 4.7b Compute the variance of the number of heads resulting from 10 indepen-
dent tosses of a fair coin.

SOLUTION Letting

Ij=

{

1 if thejth toss lands heads

0 if thejth toss lands tails