19.7. Really Great Expectations 819
temperature,T, of a cow as a random variable. Carefully specify the probability
space on whichT is defined: what are the outcomes? what are their probabilities?
Explain the precise connection between properties ofTand average herd tempera-
ture that justifies the application of Markov’s Bound.
Homework Problems
Problem 19.3.
IfRis a nonnegative random variable, then Markov’s Theorem gives an upper
bound on PrŒRxçfor any real numberx >ExŒRç. Ifbis 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.
(b)What value ofb 0 in part (a) gives the best bound?Exam Problems
Problem 19.4.
A herd of cows is stricken by an outbreak ofhot 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 19.1.1 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 aver-
ages, 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. Care-
fully specify the probability space on whichTis 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.