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Chapter 7: Q.52 (page 356)
A population is made up of disjoint subgroups. Let denote the proportion of the population that is in subgroup . If the average weight of the members of subgroup is , what is the average weight of the members of the population?
The average weight of the members of the population is
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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?
A deck of n cards numbered 1 through n is thoroughly shuffled 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.
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: Define 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?
Repeat Problem 7.68 when the proportion of the population having a value of less than is equal to .
The number of accidents that a person has in a given year is a Poisson random variable with mean. However, suppose that the value ofchanges from person to person, being equal to for percent of the population and for the otherpercent. If a person is chosen at random, what is the probability that he will have
a. We are required to find
b. We are required to find .
c. Define as the number of accidents in a preceding year. As likely as we are require to find.
Let be arbitrary events, and define
{at least of the occur}. Show that
Hint: Let denote the number of the that occur. Show
that both sides of the preceding equation are equal to .
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