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Chapter 4: Q. 4. 26 (page 172)
Prove
Hint: Use integration by parts.
The idea of the proob is integrating by parts the right side of the equation times.
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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?
Two fair dice are rolled. Let equal the product of the dice. Compute .
The expected number of typographical errors on a page of a certain magazine is. What are the probability that the next page you read contains (a) and (b) or more typographical errors? Explain your reasoning!
From a set of n elements, a nonempty subset is chosen at random in the sense that all of the nonempty subsets are equally likely to be selected. Let X denote the number of elements in the chosen subset. Using the identities given in Theoretical Exercise of Chapter, show that
Show also that for n large,
in the sense that the ratio Var(X) ton/approaches as n approaches q. Compare this formula with the limiting form of Var(Y) when P{Y =i}=/n,i=,...,n.
An urn has n white and m black balls. Balls are randomly withdrawn, without replacement, until a total of white balls have been withdrawn. The random variable equal to the total number of balls that are withdrawn is said to be a negative hypergeometric random variable.
(a) Explain how such a random variable differs from a negative binomial random variable.
(b) Find .
Hint for (b): In order for to happen, what must be the results of the first withdrawals?
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