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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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 .
Let be independent with common mean and common variance , and set . For , find
Consider the following dice game, as played at a certain gambling casino: Playersand roll a pair of dice in turn. The bank then rolls the dice to determine the outcome according to the following rule: Playerwins if his roll is strictly greater than the banks. Forlet
and show that and are positively correlated. Explain why this result was to be expected.
The joint density of and is given by
,
(a) Compute the joint moment generating function of and .
(b) Compute the individual moment generating functions.
We say that is stochastically larger than , written , if, for all ,
Show that if then when
(a) and are nonnegative random variables;
(b) and are arbitrary random variables. Hint:
Write as
where
Similarly, represent as . Then make use of part (a).
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