Chapter 17 Random Variables604
converges. Then
Ex" 1
X
iD 0RiD
X^1
iD 0ExŒRiç:Proof. LetTWWD
P 1
iD 0 Ri.
We leave it to the reader to verify that, under the given convergence hypothesis,
all the sums in the following derivation are absolutely convergent, which justifies
rearranging them as follows:
X^1iD 0ExŒRiçDX^1
iD 0X
s 2 SRi.s/PrŒsç (Def. 17.4.1)D
X
s 2 SX^1
iD 0Ri.s/PrŒsç (exchanging order of summation)D
X
s 2 S" 1
X
iD 0Ri.s/PrŒsç (factoring out PrŒsç)D
X
s 2 ST.s/PrŒsç (Def. ofT)DExŒTç (Def. 17.4.1)DEx" 1
X
iD 0Ri: (Def. ofT):17.5.6 Expectations of Products
While the expectation of a sum is the sum of the expectations, the same is usually
not true for products. For example, suppose that we roll a fair 6-sided die and
denote the outcome with the random variableR. Does ExŒRRçDExŒRçExŒRç?
We know that ExŒRçD 312 and thus ExŒRç^2 D 1214. Let’s compute ExŒR^2 çto
see if we get the same result.
ExR^2
D
X
! 2 SR^2 .!/PrŒwçDX^6
iD 1i^2 PrŒRiDiçD
12
6
C
22
6
C
32
6
C
42
6
C
52
6
C
62
6
D15 1=6¤12 1=4:
That is,
ExŒRRç¤ExŒRçExŒRç:So the expectation of a product is not always equal to the product of the expecta-
tions.