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Chapter 7: Q.7.36 (page 355)
Let be the number of and the number of that occur in rolls of a fair die. Compute .
The value ofis
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In Example 5c, compute the variance of the length of time until the miner reaches safety.
Suppose that and are independent random variables having a common mean . Suppose also that and . The value of is unknown, and it is proposed that be estimated by a weighted average of and . That is, will be used as an estimate of for some appropriate value of . Which value of yields the estimate having the lowest possible variance? Explain why it is desirable to use this value of
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.
Consider 3 trials, each having the same probability of success. Let denote the total number of successes in these trials. If , what is
(a) the largest possible value of ?
(b) the smallest possible value of }?
Let be a random variable having finite expectation and variance , and let be a twice differentiable function. Show that
Hint: Expand in a Taylor series about . Use the first
three terms and ignore the remainder.
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