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Chapter 7: Q.29 (page 361)
Let be independent and identically distributed random variables. Find
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Suppose that each of the elements of is to be colored either red or blue. Show that if are subsets of , there is a way of doing the coloring so that at most of these subsets have all their elements the same color (where denotes the number of elements in the set ).
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 }?
If 10 married couples are randomly seated at a round table, compute
(a) The expected number and
(b) The variance of the number of wives who are seated next to their husbands.
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’s 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’s 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’s T-cells (and thus has probability . of being CD4), it has dangerous consequences when facing a virus that targets only the CD4 T-cells
A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses.
(a) If you are not given any information about your earlier guesses, show that for any strategy, E[N]=1.
(b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that under this strategy
(c) Supposethatyouaretoldaftereachguesswhetheryou are right or wrong. In this case, it can be shown that the strategy that maximizes E[N] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that
Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random variables.
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