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Chapter 7: Q7.20 (page 364)
Let be a nonnegative random variable having a distribution function . Show that if , then
Hint: Start with the identity
where
if otherwise
Hence
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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.
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 an urn containing a large number of coins, and suppose that each of the coins has some probability p of turning up heads when it is flipped. However, this value of varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the value of the coin can be regarded as being the value of a random variable that is uniformly distributed over . If a coin is selected at random from the urn and flipped twice, compute the probability that
a. The first flip results in a head;
b. both flips result in heads.
Consider independent flips of a coin having probability of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if and the outcome is, then there are changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of Bernoulli random variables.
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.
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