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Chapter 7: Q. 7.51 (page 363)
Use Table to determine the distribution of when are independent and identically distributed exponential random variables, each having mean.
The identically distributed exponential random variables value are.
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Let Z be a standard normal random variable,and, for a 铿亁ed x, set
A coin having probability p of coming up heads is continually flipped until both heads and tails have appeared. Find
(a) the expected number of flips,
(b) the probability that the last flip lands on heads.
Let be independent random variables having an unknown continuous distribution function and let be independent random variables having an unknown continuous distribution function . Now order those variables, and let
The random variable is the sum of the ranks of the sample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether and are identical distributions. This test accepts the hypothesis that when is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of .
Hint: Use the results of Example 3e.
Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunter independently hits his target with probability ., compute the expected number of ducks that are hit. Assume that the number of ducks in a flock is a Poisson random variable with mean .
The county hospital is located at the center of a square whose sides are miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangular, so the travel distance from the hospital, whose coordinates are , to the point is . If an accident occurs at a point that is uniformly distributed in the square, find the expected travel distance of the ambulance.
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