Mathematics for Computer Science

17.5. Linearity of Expectation 607

for allb 2 V, then

the eventsŒRDSçandŒSDTçare independent.

This implies that these events are also independent ifTis uniform, sinceRand
T are symmetric in this assertion. Prove converssely that if neitherRnorT is
uniform, then these events are not independent.

Problems for Section 17.3

Practice Problems

Problem 17.4.
SupposeX 1 ,X 2 , andX 3 are three mutually independent random variables, each
having the uniform distribution

PrŒXiDkçequal to1=3for each ofkD1;2;3.

LetM be another random variable giving the maximum of these three random
variables. What is the density function ofM?

Class Problems

Guess the Bigger Number Game

Team 1:

 Write different integers between 0 and 7 on two pieces of paper.

 Put the papers face down on a table.

Team 2:

 Turn over one paper and look at the number on it.

 Either stick with this number or switch to the unseen other number.

Team 2 wins if it chooses the larger number.

Problem 17.5.
The analysis in section 17.3.3 implies that Team 2 has a strategy that wins 4/7 of
the time no matter how Team 1 plays. Can Team 2 do better? The answer is “no,”