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Chapter 7: Q.7.15 (page 353)
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.
The expected number of matched pairs is
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Consider a gambler who, at each gamble, either wins or loses her bet with respective probabilities and . A popular gambling system known as the Kelley strategy is to always bet the fraction of your current fortune when . Compute the expected fortune aftergambles of a gambler who starts with units and employs the Kelley strategy.
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 .
For Example , show that the variance of the number of coupons needed to a mass a full set is equal to
When is large, this can be shown to be approximately equal (in the sense that their ratio approaches 1 as ) to .
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.
A fair die is rolled times. Calculate the expected sum of the rolls.
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