Chapter 18 Deviation from the Mean648
By Total Expectation,
ExŒLçDX
i 2 ZCExŒLj1st red inith spinçPrŒ1st red inith spinçD
X
i 2 ZC.10.2iC^1 1// 2 iX
i 2 ZC10 D1:
That is, you can expect to lose an infinite amount of money before finally winning
$10 —giving you a percentage profit of 0.
So yes, probability theory leads to the absurd conclusion that, even with the odds
heavily against you, you’re certain to win playing roulette, but only if you make the
absurd assumption that you have an infinite bankroll. We can’t fault the theory for
reaching an absurd conclusion from an absurd assumption.
Problems for Section 18.2
Class Problems
Problem 18.1.
A herd of cows is stricken by an outbreak ofcold cow disease. The disease lowers
the normal body temperature of a cow, and a cow will die if its temperature goes
below 90 degrees F. The disease epidemic is so intense that it lowered the average
temperature of the herd to 85 degrees. Body temperatures as low as 70 degrees,but
no lower, were actually found in the herd.
(a)Prove that at most 3/4 of the cows could have survived.Hint:LetTbe the temperature of a random cow. Make use of Markov’s bound.
(b)Suppose there are 400 cows in the herd. Show that the bound of part (a) is best
possible by giving an example set of temperatures for the cows so that the average
herd temperature is 85, and with probability 3/4, a randomly chosen cow will have
a high enough temperature to survive.
Homework Problems
Problem 18.2.
IfRis a nonnegative random variable, then Markov’s Theorem gives an upper
bound on PrŒR xçfor any real numberx >ExŒRç. If a constantb 0 is a
lower bound onR, then Markov’s Theorem can also be applied toR bto obtain
a possibly different bound on PrŒRxç.
(a)Show that ifb > 0, applying Markov’s Theorem toR bgives a smaller
upper bound on PrŒRxçthan simply applying Markov’s Theorem directly toR.