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Chapter 4: Q. 4.1 (page 173)
Suppose that the random variable is equal to the number of hits obtained by a certain baseball player in his next at-bats. If and, find
The value ofis
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Let be a Poisson random variable with parameter . Show that increases monotonically and then decreases monotonically asincreases, reaching its maximum when is the largest integer not exceeding .
Hint: Consider .
An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X denote the selection number of the 铿乺st chosen ball that is blue. For instance, if the 铿乺st selection is red and the second blue, then X is equal to .
In the game of Two-Finger Morra, players show or fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.
(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the possibilities is equally likely, what are the possible values of and what are their associated probabilities?
(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up or fingers, what are the possible values of and their associated probabilities?
Suppose that the distribution function of X given by
(a) Find .
(b) Find .
Consider coins, each of which independently comes up heads with probability . Suppose that is large and is small, and let . Suppose that all coins are tossed; if at least one comes up heads, the experiment ends; if not, we again toss all coins, and so on. That is, we stop the first time that at least one of the coins come up heads. Let denote the total number of heads that appear. Which of the following reasonings concerned with approximating is correct (in all cases, is a Poisson random variable with parameter ?
(a) Because the total number of heads that occur when all coins are rolled is approximately a Poisson random variable with parameter ,
(b) Because the total number of heads that occur when all coins are rolled is approximately a Poisson random variable with parameter , and because we stop only when this number is positive,
(c) Because at least one coin comes up heads, will equal 1 if none of the other coins come up heads. Because the number of heads resulting from these coins is approximately Poisson with mean ,
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