Mathematics for Computer Science

Chapter 18 Random Variables772

Hint:LetA^1 WWDAandA^0 WWDA, so the eventŒIADcçis the same asAcfor
c2f0;1g; likewise forB^1 ;B^0.

Homework Problems

Problem 18.2.
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:

The probability thatRDSis the same as the probability thatRtakes

whatever valueShappens to have, therefore

PrŒRDSçD

1

jVj

: (18.17)

Are you convinced by this argument? Write out a careful proof of (18.17).

Hint:The eventŒRDSçis a disjoint union of events

ŒRDSçD

[

b 2 V

ŒRDbANDSDbç:

(b)LetSTbe the random variable giving the values ofSandT.^3 Now suppose
Rhas a uniform distribution, andRis independent ofST. How about this
argument?

The probability thatRDSis the same as the probability thatRequals

the first coordinate of whatever valueSThappens to have, and this

probability remains equal to1=jVjby independence. Therefore the

eventŒRDSçis independent ofŒSDTç.

Write out a careful proof thatŒRDSçis independent ofŒSDTç.

(^3) That is,STWS!VVwhere
.ST/.!/WWD.S.!/;T.!//
for every outcome! 2 S.