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Chapter 7: Q.7.45 (page 362)
For a standard normal random variable ,
Show that
Hint: Start by expanding the moment generating function of into a Taylor series about to obtain
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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.
Two envelopes, each containing a check, are placed in front of you. You are to choose one of the envelopes, open it, and see the amount of the check. At this point, either you can accept that amount or you can exchange it for the check in the unopened envelope. What should you do? Is it possible to devise a strategy that does better than just accepting the first envelope? Let and , , denote the (unknown) amounts of the checks and note that the strategy that randomly selects an envelope and always accepts its check has an expected return of . Consider the following strategy: Let be any strictly increasing (that is, continuous) distribution function. Choose an envelope randomly and open it. If the discovered check has the value , then accept it with probability and exchange it with probability .
(a) Show that if you employ the latter strategy, then your expected return is greater than . Hint: Condition on whether the first envelope has the value or . Now consider the strategy that fixes a value x and then accepts the first check if its value is greater than x and exchanges it otherwise. (b) Show that for any , the expected return under the-strategy is always at least and that it is strictly larger than if lies between and .
(c) Let be a continuous random variable on the whole line, and consider the following strategy: Generate the value of, and if , then employ the -strategy of part (b). Show that the expected return under this strategy is greater than .
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 -of--out-of- circular reliability system, , consists of components that are arranged in a circular fashion. Each component is either functional or failed, and the system functions if there is no block of consecutive components of which at least are failed. Show that there is no way to arrange components, of which are failed, to make a functional -of--out-of-circular system.
For a group of 100 people, compute
(a) the expected number of days of the year that are birthdays of exactly 3 people;
(b) the expected number of distinct birthdays.
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