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Chapter 7: Q. 7.4 (page 359)
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.
The random variable is showed as having finite expectation.
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A group of men and women is lined up at random.
(a) Find the expected number of men who have a woman next to them.
(b) Repeat part (a), but now assuming that the group is randomly seated at a round table.
The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the object of the game is to guess the chosen three.
(a) How many solutions are possible? In one version of the game, the selection is made and then each of the players is randomly given three of the remaining cards. Let S, W, and R be, respectively, the numbers of suspects, weapons, and rooms in the set of three cards given to a specified player. Also, let X denote the number of solutions that are possible after that player observes his or her three cards.
(b) Express X in terms of S, W, and R.
(c) Find E[X]
Let be the standard normal distribution function, and let X be a normal random variable with mean 渭 and variance 1. We want to find E[ (X)]. To do so, let Z be a standard normal random variable that is independent of X, and let
(a) Show that .
(b) Show that .
(c) Show that .
Hint: What is the distribution of ?
The preceding comes up in statistics. Suppose you are about to observe the value of a random variable X that is normally distributed with an unknown mean 渭 and variance 1, and suppose that you want to test the hypothesis that the mean 渭 is greater than or equal to 0. Clearly you would want to reject this hypothesis if X is sufficiently small. If it results that X = x, then the p-value of the hypothesis that the mean is greater than or equal to 0 is defined to be the probability that X would be as small as x if 渭 were equal to 0 (its smallest possible value if the hypothesis were true). (A small p-value is taken as an indication that the hypothesis is probably false.) Because X has a standard normal distribution when 渭 = 0, the p-value that results when X = x is (x). Therefore, the preceding shows that the expected p-value that results when the true mean is 渭 is .
Suppose that A and B each randomly and independently chooseofobjects. Find the expected number of objects
a. Chosen by both A and B;
b. Not chosen by either A or B;
c. Chosen by exactly one of A and B.
Consider a population consisting of individuals able to produce offspring of the same kind. Suppose that by the end of its lifetime, each individual will have produced j new offspring with probability Pj, , independently of the number produced by any other individual. The number of individuals initially present, denoted by X0, is called the size of the zeroth generation. All offspring of the zeroth generation constitute the first generation, and their number is denoted by X1. In general, let Xn denote the size of the nth generation. Let and denote, respectively, the mean and the variance of the number of offspring produced by a single individual. Suppose that X0 = 1鈥� that is, initially there is a single individual in the population
(a) Show that .
(b) Use part (a) to conclude that
(c) Show that
(d) Use part (c) to conclude that
The model just described is known as a branching process, and an important question for a population that evolves along such lines is the probability that the population will eventually die out. Let 蟺 denote this probability when the population starts with a single individual. That is,
(e) Argue that 蟺 satisfies
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