Mathematics for Computer Science

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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:

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