Introduction to Probability and Statistics for Engineers and Scientists

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 }