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Chapter 7: Q.7.3 (page 354)
If and Y are independent and identically distributed with mean and variance , find
The value of 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.
Urn contains white and black balls, while urn contains white and black balls. Two balls are randomly selected from urn and are put into urn . If balls are then randomly selected from urn , compute the expected number of white balls in the trio.
Hint: LetXi = if the i th white ball initially in urn is one of the three selected, and let Xi = otherwise. Similarly, let Yi = if the i the white ball from urn is one of the three selected, and let Yi = otherwise. The number of white balls in the trio can now be written as
Show that is minimized at .
Let be the number of and the number of that occur in rolls of a fair die. Compute .
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 .
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