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Chapter 7: Q.7.76 (page 358)
Let be the value of the first die and the sum of the values when two dice are rolled. Compute the joint moment generating function of and .
The joint moment generating functions ofandare.
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Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain
(a) 2 aces;
(b) 5 spades;
(c) all 13 hearts.
Consider independent trials, the of which results in a success with probability .
(a) Compute the expected number of successes in the trials-call it
(b) For a fixed value of , what choice of maximizes the variance of the number of successes?
(c) What choice minimizes the variance?
In Example h,say that i and, form a matched pair if i chooses the hat belonging to j and j chooses the hat belonging to i. Find the expected number of matched pairs.
A deck of n cards numbered 1 through n is thoroughly shuf铿俥d so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses.
(a) If you are not given any information about your earlier guesses, show that for any strategy, E[N]=1.
(b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that under this strategy
(c) Supposethatyouaretoldaftereachguesswhetheryou are right or wrong. In this case, it can be shown that the strategy that maximizes E[N] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that
Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random variables.
Show that is stochastically larger than if and only if
for all increasing functions .
Hint: Show that , then by showing that and then using Theoretical Exercise 7.7. To show that if for all increasing functions , then , define an appropriate increasing function .
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