Introduction to Probability and Statistics for Engineers and Scientists

4.7Covariance and Variance of Sums of Random Variables 123

Proof

Cov





∑n

i= 1

Xi,

∑m

j= 1

Yj





=

∑n

i= 1

Cov



Xi,

∑m

j= 1

Yj



 from Equation 4.7.5

=

∑n

i= 1

Cov





∑m

j= 1

Yj,Xi



 by the symmetry property Equation 4.7.2

=

∑n

i= 1

∑m

j= 1

Cov(Yj,Xi) again from Equation 4.7.5

and the result now follows by again applying the property Equation 4.7.2.

Using Equation 4.7.3 gives rise to the following formula for the variance of a sum of
random variables.

Corollary 4.7.3

Var

( n

∑

i= 1

Xi

)

=

∑n

i= 1

Var(Xi)+

∑n

i= 1

∑n

j= 1

j=i

Cov(Xi,Xj)

Proof

The proof follows directly from Proposition 4.7.2 upon settingm=n, andYj=Xjfor
j=1,...,n.

In the case ofn=2, Corollary 4.7.3 yields that

Var(X+Y)=Var(X)+Var(Y)+Cov(X,Y)+Cov(Y,X)

or, using Equation 4.7.2,

Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y) (4.7.6)

Theorem 4.7.4

IfXandYare independent random variables, then

Cov(X,Y)= 0