17.5. Linearity of Expectation 605
There is a special case when such a relationshipdoeshold however; namely,
when the random variables in the product areindependent.
Theorem 17.5.6.For any twoindependentrandom variablesR 1 ,R 2 ,
ExŒR 1 R 2 çDExŒR 1 çExŒR 2 ç:
The proof follows by judicious rearrangement of terms in the sum that defines
ExŒR 1 R 2 ç. Details appear in Problem 17.15.
Theorem 17.5.6 extends routinely to a collection of mutually independent vari-
ables.
Corollary 17.5.7.[Expectation of Independent Product] If random variablesR 1 ;R 2 ;:::;Rk
are mutually independent, then
ExŒ
Yk
iD 1
RiçD
Yk
iD 1
ExŒRiç:
Problems for Section 17.2
Practice Problems
Problem 17.1. (a)Prove that ifAandBare independent events, then so areAand
B.
(b)LetIAandIBbe the indicator variables for eventsAandB. Prove thatIA
andIBare independent iffAandBare independent.
Hint:For any event,E, letE^1 WWDEandE^0 WWDE. So the eventŒIEDaçis the
same asEa.
Homework Problems
Problem 17.2.
LetR,S, andTbe random variables with the same codomain,V.
(a)SupposeRis uniform —that is,
PrŒRDbçD
1
jVj
;
for allb 2 V —andRis independent ofS. Originally this text had the following
argument: