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