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Chapter 7: Q.7.12 (page 353)
A set of cards numbered 1 through is randomly distributed among people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card.
One of the people expected to receive a card with a number that matches its age.
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Consider the following dice game: A pair of dice is rolled. If the sum isthen the game ends and you win If the sum is not then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value of find your expected return if you employ the strategy of stopping the first time that a value at least as large as appears. What value ofleads to the largest expected return? Hint: Let denote the return when you use the critical value To compute, condition on the initial sum.
The number of people who enter an elevator on the ground floor is a Poisson random variable with mean . If there are floors above the ground floor, and if each person is equally likely to get off at any one of the floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.
Let be a sequence of independent and identically distributed continuous random variables. Let be such that
That is, is the point at which the sequence stops decreasing. Show that .
Hint: First find .
Letbe a sequence of independent uniformrandom variables. In Example , we showed that for , where
This problem gives another approach to establishing that result.
(a) Show by induction on n that for 0 and all
Hint: First condition onand then use the induction hypothesis.
use part (a) to conclude that
A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.
(a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].
Hint: De铿乶e n + m indicator variables, one for each of the small pills initially present and one for each of the small pills created when a large one is split in two. Now use the argument of Example m.
(b) Let Y denote the day on which the last large pills chosen. Find E[Y].
Hint: What is the relationship between X and Y?
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