Mathematics for Computer Science

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 ç
ExŒR 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: