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Chapter 4: Q.4.73 (page 168)
Suppose in Problem 4.72 that the two teams are evenly matched and each has probability 1 2 of winning each game. Find the expected number of games played.
In the given information the answer is
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Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability , then he or she will receive a score of
We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability , what value of should he or she assert so as to maximize the expected score?
Suppose that the number of events that occur in a specified time is a Poisson random variable with parameter . If each event is counted with probability , independently of every other event, show that the number of events that are counted is a Poisson random variable with parameter . Also, give an intuitive argument as to why this should be so. As an application of the preceding result, suppose that the number of distinct uranium deposits in a given area is a Poisson random variable with parameter . If, in a fixed period of time, each deposit is discovered independently with probability , find the probability that
(a) exactly ,
(b) at least , and
(c) at most deposit is discovered during that time.
The random variable X is said to have the Yule-Simons distribution if
(a) Show that the preceding is actually a probability mass function. That is, show that
(b) Show that E[X] = 2.
(c) Show that E[X2] = q
Consider a roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. If Smith always bets that the outcome will be one of the numbers 1 through 12, what is the probability that
If you buy a lottery ticket in lotteries, in each of which your chance of winning a prize is role="math" localid="1646465220038" , what is the (approximate) probability that you will win a prize
(a) at least once?
(b) exactly once?
(c) at least twice?
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