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