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Chapter 7: Q. 7.10 (page 364)
Let be a Poisson random variable with mean . Show that if is not too small, then
Hint: Use the result of Theoretical Exercise to approximate.
If is not too small, then is shown.
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Show that is minimized at .
If items are distributed among boxes, then at least one of the boxes must contain more than items. Use the probabilistic method to prove this result.
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.
N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the pairs of people is, independently, a pair of friends with probability p, find the expected number of occupied tables.
Hint: Let equal or , depending on whether theth arrival sits at a previously unoccupied table.
Let be independent and identically distributed positive random variables. For find
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