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Chapter 7: Q. 7.27 (page 354)
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.
Yes, there is a box containing items.
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A certain region is inhabited by r distinct types of a certain species of insect. Each insect caught will, independently of the types of the previous catches, be of type i with probability
(a) Compute the mean number of insects that are caught before the 铿乺st type catch.
(b) Compute the mean number of types of insects that are caught before the 铿乺st type catch.
Cards from an ordinary deck of playing cards are turned face upon at a time. If the 1st card is an ace, or the nd a deuce, or the rd a three, or ...,or the th a king,or the an ace, and so on, we say that a match occurs. Note that we do not require that the (n + ) card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.
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.
The positive random variable is said to be a lognormal random variable with parameters and if is a normal random variable with mean and variance role="math" localid="1647407606488" . Use the normal moment generating function to find the mean and variance of a lognormal random variable
A fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 6 and a 5. Find
(a) ;
(b) ;
(c) ;
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