Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 883 — #891
20.7. Really Great Expectations 883
bound on the deviation of a random variable. Justify this approach by regarding
the temperature T of a cow as a random variable. Carefully specify the probability
space on which T is defined: what are the sample points? what are their probabilities?
Explain the precise connection between properties of T and average herd
temperature that justifies the application of Markov’s Bound.
Homework Problems
Problem 20.3.
If R is a nonnegative random variable, then Markov’s Theorem gives an upper
bound on PrŒR x for any real number x > ExŒR. If b is a lower bound on R,
then Markov’s Theorem can also be applied to R b to obtain a possibly different
bound on PrŒR x.
(a) Show that if b > 0, applying Markov’s Theorem to R b gives a smaller
upper bound on PrŒR x than simply applying Markov’s Theorem directly to R.
(b) What value of b 0 in part (a) gives the best bound?
Exam Problems
Problem 20.4.
A herd of cows is stricken by an outbreak of hot cow disease. The disease raises
the normal body temperature of a cow, and a cow will die if its temperature goes
above 90 degrees. The disease epidemic is so intense that it raised the average
temperature of the herd to 120 degrees. Body temperatures as high as 140 degrees,
but no higher, were actually found in the herd.
(a) Use Markov’s Bound 20.1.1 to prove that at most 2/5 of the cows could have
survived.
(b) Notice that the conclusion of part (a) is a purely arithmetic facts about averages,
not about probabilities. But you verified the claim of part (a) by applying
Markov’s bound on the deviation of a random variable. Justify this approach by
explaining how to define a random variable T for the temperature of a cow. Carefully
specify the probability space on which T is defined: what are the outcomes?
what are their probabilities? Explain the precise connection between properties of
T , average herd temperature, and fractions of the herd with various temperatures,
that justify application of Markov’s Bound.