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Chapter 4: Q.4.19 (page 171)
Show that is a Poisson random variable with parameter , then
Now use this result to compute .
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Suppose that takes on one of the valuesand. If for some constant, find
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?
A communications channel transmits the digits and However, due to static, the digit transmitted is incorrectly received with probability Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit instead of and 11111 instead of If the receiver of the message uses 鈥渕ajority鈥� decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?
Letbe the winnings of a gambler. Let and suppose that
Compute the conditional probability that the gambler wins given that he wins a positive amount.
To determine whether they have a certain disease, people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to place the people into groups of . The blood samples of the people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the people, whereas if the test is positive, each of the people will also be individually tested and, in all, tests will be made on this group. Assume that the probability that a person has the disease isrole="math" localid="1646542351988" for all people, independently of one another, and compute the expected number of tests necessary for each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.)
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