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Chapter 7: Q.7.1 (page 359)
Show that is minimized at .
Differentiaterespective to.
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The number of winter storms in a good year is a Poisson random variable with a mean of , whereas the number in a bad year is a Poisson random variable with a mean of. If next year will be a good year with probability .or a bad year with probability , find the expected value and variance of the number of storms that will occur.
A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of married couples that are paired together.
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 number of people who enter an elevator on the ground floor is a Poisson random variable with mean . If there are floors above the ground floor, and if each person is equally likely to get off at any one of the floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.
An urn has black balls. At each stage, a black ball is removed and a new ball that is black with probability and white with probability is put in its place. Find the expected number of stages needed until there are no more black balls in the urn. note: The preceding has possible applications to understanding the AIDS disease. Part of the body鈥檚 immune system consists of a certain class of cells known as T-cells. There are types of T-cells, called CD4 and CD8. Now, while the total number of T-cells in AIDS sufferers is (at least in the early stages of the disease) the same as that in healthy individuals, it has recently been discovered that the mix of CD4 and CD8 T-cells is different. Roughly 60 percent of the T-cells of a healthy person are of the CD4 type, whereas the percentage of the T-cells that are of CD4 type appears to decrease continually in AIDS sufferers. A recent model proposes that the HIV virus (the virus that causes AIDS) attacks CD4 cells and that the body鈥檚 mechanism for replacing killed T-cells does not differentiate between whether the killed T-cell was CD4 or CD8. Instead, it just produces a new T-cell that is CD4 with probability . and CD8 with probability .. However, although this would seem to be a very efficient way of replacing killed T-cells when each one killed is equally likely to be any of the body鈥檚 T-cells (and thus has probability . of being CD4), it has dangerous consequences when facing a virus that targets only the CD4 T-cells
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