2.6. Extension to Several Random Variables 141ThenE[A 1 W 1 +A 2 W 2 ]=A 1 E[W 1 ]+A 2 E[W 2 ] (2.6.11)
E[A 1 W 1 B]=A 1 E[W 1 ]B. (2.6.12)Proof:Because of the linearity of the operatorEon random variables, we have for
the (i, j)th components of expression (2.6.11) that
E[m
∑s=1a 1 isW 1 sj+∑ms=1a 2 isW 2 sj]
=∑ms=1a 1 isE[W 1 sj]+∑ms=1a 2 isE[W 2 sj].Hence by (2.6.10), expression (2.6.11) is true. The derivation of expression (2.6.12)
follows in the same manner.LetX=(X 1 ,...,Xn)′be ann-dimensional random vector, such thatσ^2 i =
Var(Xi)<∞.ThemeanofXisμ=E[X] and we define itsvariance-covariance
matrixas
Cov(X)=E[(X−μ)(X−μ)′]=[σij], (2.6.13)whereσiidenotesσ^2 i. As Exercise 2.6.8 shows, theith diagonal entry of Cov(X)is
σi^2 =Var(Xi)andthe(i, j)th off diagonal entry is Cov(Xi,Xj).
Example 2.6.3(Example 2.5.6, Continued).In Example 2.5.6, we considered
the joint pdf
f(x, y)={
e−y 0 <x<y<∞
0elsewhere,and showed that the first two moments areμ 1 =1,μ 2 =2
σ^21 =1,σ^22 = 2 (2.6.14)
E[(X−μ 1 )(Y−μ 2 )] = 1.LetZ=(X, Y)′. Then using the present notation, we have
E[Z]=[
1
2]
and Cov(Z)=[
11
12]
.Two properties of Cov(Xi,Xj) needed later are summarized in the following
theorem:
Theorem 2.6.3. LetX=(X 1 ,...,Xn)′be ann-dimensional random vector, such
thatσ^2 i=σii=Var(Xi)<∞.LetAbe anm×nmatrix of constants. Then
Cov(X)=E[XX′]−μμ′ (2.6.15)
Cov(AX)=ACov(X)A′. (2.6.16)