4.7Covariance and Variance of Sums of Random Variables 125
then the total number of heads is equal to
∑^10
j= 1
Ij
Hence, from Theorem 4.7.4,
Var
∑^10
j= 1
Ij
=
∑^10
j= 1
Var(Ij)
Now, sinceIjis an indicator random variable for an event having probability^12 , it follows
from Example 4.6b that
Var(Ij)=^12
(
1 −^12
)
=^14
and thus
Var
∑^10
j= 1
Ij
=^10
4
■
The covariance of two random variables is important as an indicator of the relationship
between them. For instance, consider the situation whereXandYare indicator variables
for whether or not the eventsAandBoccur. That is, for eventsAandB, define
X=
{
1ifAoccurs
0 otherwise
, Y=
{
1ifBoccurs
0 otherwise
and note that
XY=
{
1ifX=1,Y= 1
0 otherwise
Thus,
Cov(X,Y)=E[XY]−E[X]E[Y]
=P{X=1,Y= 1 }−P{X= 1 }P{Y= 1 }
From this we see that
Cov(X,Y)> 0 ⇔P{X=1,Y= 1 }>P{X= 1 }P{Y= 1 }