Mathematics for Computer Science

Chapter 17 Random Variables606

The probability thatRDSis the same as the probability thatRtakes

whatever valueShappens to have, therefore

PrŒRDSçD

1

jVj

: (17.16)

Are you convinced by this argument? Write out a careful proof of (17.16).

Hint:The eventŒRDSçis a disjoint union of events

ŒRDSçD

[

b 2 V

ŒRDbANDSDbç:

(b)LetSTbe the random variable giving the values ofSandT.^5 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.

(c)LetV Df1;2;3gandR;S;Ttake the following values with equal probability,

111;211;123;223;132;232:

Verify that

1.Ris independent ofST,

2. The eventŒRDSçis not independent ofŒSDTç.
   3.SandThave a uniform distribution,

Problem 17.3.
LetR,S, andTbe mutually independent random variables with the same codomain,
V. Problem 17.2 showed that ifRis uniform —that is,

PrŒRDbçD

1

jVj

;

(^5) That is,STWS!VVwhere
.ST/.!/WWD.S.!/;T.!//
for every outcome! 2 S.