Chapter 7

The number of winter storms in a good year is a Poisson random variable with mean \(3,\) whereas the number in a bad year is a Poisson random variable with mean \(5 .\) If next year will be a good year with probability .4 or a bad year with probability.6, find the expected value and variance of the number of storms that will occur.

In Example \(5 \mathrm{c},\) compute the variance of the length of time until the miner reaches safety.

Consider a gambler who, at each gamble, either wins or loses her bet with respective probabilities \(p\) and \(1-p\) A popular gambling system known as the Kelley strategy is to always bet the fraction \(2 p-1\) of your current fortune when \(p>\frac{1}{2} .\) Compute the expected fortune after \(n\) gambles of a gambler who starts with \(x\) units and employs the Kelley strategy.

The number of accidents that a person has in a given year is a Poisson random variable with mean \(\lambda .\) However, suppose that the value of \(\lambda\) changes from person to person, being equal to 2 for 60 percent of the population and 3 for the other 40 percent. If a person is chosen at random, what is the probability that he will have (a) 0 accidents and (b) exactly 3 accidents in a certain year? What is the conditional probability that he will have 3 accidents in a given year, given that he had no accidents the preceding year?

Repeat Problem 7.68 when the proportion of the population having a value of \(\lambda\) less than \(x\) is equal to \(1-e^{-x}\)

Consider an urn containing a large number of coins, and suppose that each of the coins has some probability \(p\) of turning up heads when it is flipped. However, this value of \(p\) varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the \(p\) -value of the coin can be regarded as being the value of a random variable that is uniformly distributed over \([0,1] .\) If a coin is selected at random from the urn and flipped twice, compute the probability that (a) the first flip results in a head; (b) both flips result in heads.

In Example \(6 \mathrm{c},\) suppose that \(X\) is uniformly distributed over \((0,1) .\) If the discretized regions are determined by \(a_{0}=0, a_{1}=\frac{1}{2},\) and \(a_{2}=1,\) calculate the optimal quantizer \(Y\) and compute \(E\left[(X-Y)^{2}\right]\)

The moment generating function of \(X\) is given by \(M_{X}(t)=\exp \left\\{2 e^{t}-2\right\\}\) and that of \(Y\) by \(M_{Y}(t)=\) \(\left(\frac{3}{4} e^{t}+\frac{1}{4}\right)^{10} .\) If \(X\) and \(Y\) are independent, what are (a) \(P\\{X+Y=2\\} ?\) (b) \(P\\{X Y=0\\} ?\) (c) \(E[X Y] ?\)

Let \(X\) be the value of the first die and \(Y\) the sum of the values when two dice are rolled. Compute the joint moment generating function of \(X\) and \(Y\).

The joint density of \(X\) and \(Y\) is given by \(f(x, y)=\frac{1}{\sqrt{2 \pi}} e^{-y} e^{-(x-y)^{2} / 2} \quad 0

Two envelopes, each containing a check, are placed in front of you. You are to choose one of the envelopes, open it, and see the amount of the check. At this point, either you can accept that amount or you can exchange it for the check in the unopened envelope. What should you do? Is it possible to devise a strategy that does better than just accepting the first envelope? Let \(A\) and \(B, A

Successive weekly sales, in units of \(\$ 1,000,\) have a bivariate normal distribution with common mean \(40,\) common standard deviation \(6,\) and correlation. 6 (a) Find the probability that the total of the next 2 weeks' sales exceeds 90. (b) If the correlation were .2 rather than \(.6,\) do you think that this would increase or decrease the answer to (a)? Explain your reasoning. (c) Repeat (a) when the correlation is . 2