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Chapter 7: Q.7.22 (page 354)
How many times would you expect to roll a fair die before all sides appeared at least once?
The expected number is
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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 function ofandis given by
Find and show that
The joint density of and is given by
Compute .
Suppose that and are independent random variables having a common mean . Suppose also that and . The value of is unknown, and it is proposed that be estimated by a weighted average of and . That is, will be used as an estimate of for some appropriate value of . Which value of yields the estimate having the lowest possible variance? Explain why it is desirable to use this value of
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.
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