Introduction to Probability and Statistics for Engineers and Scientists

122 Chapter 4:Random Variables and Expectation

=E[XY]−μxμy−μyμx+μxμy

=E[XY]−E[X]E[Y] (4.7.1)

From its definition we see that covariance satisfies the following properties:

Cov(X,Y)=Cov(Y,X) (4.7.2)

and

Cov(X,X)=Var(X) (4.7.3)

Another property of covariance, which immediately follows from its definition, is that, for
any constanta,

Cov(aX,Y)=aCov(X,Y) (4.7.4)

The proof of Equation 4.7.4 is left as an exercise.
Covariance, like expectation, possesses an additive property.

Lemma 4.7.1

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

Proof

Cov(X+Z,Y)

=E[(X+Z)Y]−E[X+Z]E[Y] from Equation 4.7.1

=E[XY]+E[ZY]−(E[X]+E[Z])E[Y]

=E[XY]−E[X]E[Y]+E[ZY]−E[Z]E[Y]

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

Lemma 4.7.1 can be easily generalized (see Problem 48) to show that

Cov

(n

∑

i= 1

Xi,Y

)

=

∑n

i= 1

Cov(Xi,Y) (4.7.5)

which gives rise to the following.

PROPOSITION 4.7.2

Cov





∑n

i= 1

Xi,

∑m

j= 1

Yj



=

∑n

i= 1

∑m

j= 1

Cov(Xi,Yj)