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mathematics
a first course in probability

A First Course In Probability 10th Edition Sheldon Ross - Solutions

Show that if X and Y are independent, thenE[X| Y = y] = E[X] for all ya. In the discrete case;b. In the continuous case.
Prove that [E(g(X)Y|X] = g(X)E[Y|X].
Prove that if E[Y|X = x] = E[Y] for all x, then and are uncorrelated; give a counterexample to show that the converse is not true.
For an event A, let IA equal 1 if occurs and let it equal 0 if A does not occur. For a random variable X, show that E[XI ] P(A) E[X|A]
A coin that lands on heads with probability p is continually flipped. Compute the expected number of flips that are made until a string of r heads in a row is obtained. E[X] > pi-1(i+ E[X]) %3D i = 1 Σ +(1- p) pi-lr i =r +1
For another approach to Theoretical Exercise 7.34, let TT denote the number of flips required to obtain a run of consecutive heads.a. Determine E[TT|TT-1].b. Determine E[TT] in terms of E[Tr-1].c. What is E[T1]?d. What is E[TT]? E[X] > pi-1(i+ E[X]) %3D i = 1 Σ +(1- p) pi-lr i =r +1
The probability generating function of the discrete nonnegative integer valued random variable having probability mass function pj, j ≥ 0, is defined byLet be a geometric random variable with parameter p = 1 - s, where 0 < s < 1. Suppose that is independent of X, and show thatϕ(s) = P{X
Suppose that 2 balls are randomly removed from an urn containing n red and m blue balls. Let Xi = 1 if the ith ball removed is red, and let it be otherwise, i = 1, 2.a. Do you think that Cov(X1, X2) is negative, zero, or positive.b. Validate your answer to part (a). Suppose the red balls are
The best linear predictor of Y with respect to X1 and X2 is equal to a + bX1 + cX2, where a, b, and are chosen to minimizeE[(Y - (a + bX1 + cX2))2]Determine a, b, and c.
The best quadratic predictor of with respect to X is a + bX + cX2, where a, b, and c are chosen to minimize E[(Y - (a + bX + cX2))2]. Determine a, b, and c.
Suppose that X1, ... , Xn have a multivariate normal distribution. Show that X1, ... , Xn are independent random variables if and only ifCov(Xi, Xj) = 0 when i ≠ j
Suppose that the expected number of accidents per week at an industrial plant is 5. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2.5. If the number of workers injured in each accident is independent of the number of
A coin having probability p of coming up heads is continually flipped until both heads and tails have appeared. Finda. The expected number of flips;b. The probability that the last flip lands on heads.
A coin that comes up heads with probability is continually flipped. Let N be the number of flips until there have been both at least heads and at least m tails. Derive an expression for E[N] by conditioning on the number of heads in the first n + m flips.
There are n + 1 participants in a game. Each person independently is a winner with probability p. The winners share a total prize of 1 unit. (For instance, if 4 people win, then each of them receives 1/4, whereas if there are no winners, then none of the participants receives anything.) Let A
The number of goals that J scores in soccer games that her team wins is Poisson distributed with mean while the number she scores in games that her team loses is Poisson distributed with mean 1. Assume that, independent of earlier results, J’s team wins each new game it plays with probability
If the level of infection of a tree is x, then each treatment will independently be successful with probability 1 - x. Consider a tree whose infection level is assumed to be the value of a uniform (0, 1) random variable.a. Find the probability that a single treatment will result in a cure.b. Find
If has variance then σ2, the positive square root of the variance, is called the standard deviation. If has mean and standard deviation show that P{|X – µl 2 ko}< k 11/2
Suppose that is a random variable with mean and variance both equal to 20. What can be said about P{0 < X < 40}?
Suppose that a fair die is rolled 100 times. Let Xi be the value obtained on the ith roll. Compute an approximation for 100 P X sa 100 1< a< 6
Explain why a gamma random variable with parameters (t, λ) has an approximately normal distribution when t is large.
Suppose a fair coin is tossed 1000 times. If the first 100 tosses all result in heads, what proportion of heads would you expect on the final 900 tosses? Comment on the statement “The strong law of large numbers swamps but does not compensate.”
Many people believe that the daily change of price of a company’s stock on the stock market is a random variable with mean 0 and variance σ2. That is, if Yn represents the price of the stock on the nth day, thenYn = Yn-1 + Xn n ≥ 1where X1, X2, ... are independent and identically distributed
The Chernoff bound on a standard normal random variable gives P{Z > a} ≤ e-a2/2, a > 0. Show, by considering the density of that the right side of the inequality can be reduced by the factor 2. That is, show that 1 P{Z > a} 0
Show that if E[X] < 0 and θ ≠ 0 is such that E[eθX] = 1. then θ > 0.
A lake contains 4 distinct types of fish. Suppose that each fish caught is equally likely to be any one of these types. Let denote the number of fish that need be caught to obtain at least one of each type.a. Give an interval (a, b) such that P{a ≤ Y ≤ b} ≥ .90.b. Using the one-sided
If X is a nonnegative random variable with mean 25, what can be said abouta. E[X3]b. E[√X]?c. [log X]?d. E[e-X]?
Let be a nonnegative random variable. Prove that 1/2 1/3 E[X] < (E[X*]) < (E[X*I)
Would the results of Example 5f change if the investor were allowed to divide her money and invest the fraction α, 0 < α < 1, in the risky proposition and invest the remainder in the risk-free venture? Her return for such a split investment would be R = αX + (1 - α)m.
If X is a Poisson random variable with mean 100, then P{X > 120} is approximatelya. .02,b. .5 orc. .3?
Suppose that the distribution of earnings of members of a population is Pareto with parameters λ a > 0, wherea. Show that the top 20 percent of earners earn 80 percent of the total earnings.b. Show that the 20 top percent of the top 20 percent of earners earn 80 percent of the earnings of the
If L(p) is the Lorenz curve associated with the random variable X, show that |X d. L(p) E[X]
If f(x) is an increasing g(x) and is a decreasing function, show that E[f(X)g(X) ≤ E[f(X)]E[(g(X)].
Suppose that L(p) is the Lorenz curve associated with the random variable and that c > 0.a. Find the Lorenz curve associated with the random variable cX.b. Show that Lc(p), the Lorenz curve associated with the random variable X + c, isc. Verify that the answer to part (b) is in accordance
Suppose that 3 white and 3 black balls are distributed in two urns in such a way that each urn contains 3 balls. We say that the system is in state i if the first urn contains i white balls, i = 0, 1, 2, 3. At each stage, 1 ball is drawn from each urn and the ball drawn from the first urn is placed
Customers arrive at a bank at a Poisson rate Suppose that two customers arrived during the first hour. What is the probability thata. Both arrived during the first 20 minutes?b. At least one arrived during the first 20 minutes?
A certain person goes for a run each morning. When he leaves his house for his run, he is equally likely to go out either the front or the back door, and similarly, when he returns, he is equally likely to go to either the front or the back door. The runner owns 5 pairs of running shoes, which he
Determine the entropy of the sum that is obtained when a pair of fair dice is rolled.
Prove that if can take on any of possible values with respective probabilities P1, ..., Pn, then H(X) is maximized when Pi = 1/n, i = 1, ... , n. What is H(X) equal to in this case?
A pair of fair dice is rolled. Letand let equal the value of the first die. Compute(a) H(Y),(b) HY(X). and(c) H(X, Y). (1 if the sum of the dice is 6 X = 0 otherwise
A coin having probability p = 2/3 of coming up heads is flipped 6 times. Compute the entropy of the outcome of this experiment.
A random variable can take on any of possible values x1, ... , xn with respective probabilities p(xi), i = 1, ..., n. We shall attempt to determine the value of by asking a series of questions, each of which can be answered “yes” or “no.” For instance, we may ask “Is X = x1?” or “Is X
Show that for any discrete random variable and function f,H(f(X)) ≤ H(X)
Suppose it is relatively easy to simulate from for Fi each I = 1, …, n. How can we simulate froma.b. F(x) = || F:(X)? i = 1
Suppose we have a method for simulating random variables from the distributions F1 and F2. Explain how to simulate from the distribution.Give a method for simulating from F(x) = pF1(x) + (1– p)F2(x) 0
Let (X, Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thusLet R = (X2 + Y2)1/2 and θ = tan-1(Y/X) denote the polar coordinates of (X, Y). Show that R and θ are independent, with R2 being uniform on (0, 1) and θ being uniform on (0, 2π). 1
Let X be a random variable on (0, 1) whose density is f(x). Show that we can estimate f10g(x)dx by simulating X and then taking g(X)/f (X) as our estimate. This method, called importance sampling, tries to choose f similar in shape to g, so that g(X)/f(X) has a small variance.
If is uniformly distributed over (a, b), what random variable, having a linear relation with X, is uniformly distributed over (0, 1)?
Players A, B, C, D are randomly lined up. The first two players in line then play a game; the winner of that game then plays a game with the person who is third in line; the winner of that game then plays a game with the person who is fourth in line. The winner of that last game is considered the
If X has distribution function F, what is the distribution function of the random variable aX + β, where α and β are constants, α ≠ 0?
Suppose that balls are randomly distributed into compartments. Find the probability that m balls will fall into the first compartment. Assume that all Nn arrangements are equally likely.
Suppose that a nonmathematical, but philosophically minded, friend of yours claims that Laplace’s rule of succession must be incorrect because it can lead to ridiculous conclusions. “For instance,” says he, “the rule states that if a boy is 10 years old, having lived 10 years, the boy has
Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Notwanting to spoil a weekend family event, she does not want to hear any bad news in the next few days. But if she tells the doctor to call only if the news is good, then if the doctor does not call, Ms. Aquina can conclude that the
A family has children with probability pj, where p1 = .1, p2 = .25, p3 = .35, p4 = .3. A child from this family is randomly chosen. Given that this child is the eldest child in the family, find the conditional probability that the family hasa. Only child;b. Children.Redo (a) and (b) when the
In Example 3f, suppose that the new evidence is subject to different possible interpretations and in fact shows only that it is 90 percent likely that the criminal possesses the characteristic in question. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he
There is a 30 percent chance that can fix her busted computer. If cannot, then there is a 40 percent chance that her friend B can fix it.a. Find the probability it will be fixed by either A or B.b. If it is fixed, what is the probability it will be fixed by B.
a. A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin?b. Suppose that he flips the same coin a second time and, again, it shows heads. Now what is the probability
Twelve percent of all U.S. households are in California. A total of 1.3 percent of all U.S. households earn more than $250,000 per year, while a total of 3.3 percent of all California households earn more than $250,000 per year.a. What proportion of all non-California households earn more than
Players 1, 2, 3 are playing a tournament. Two of these three players are randomly chosen to play a game in round one, with the winner then playing the remaining player in round two. The winner of round two is the tournament victor. Assume that all games are independent and that wins when playing
Suppose there are two coins, with coin landing heads when flipped with probability .3 and coin with probability .5. Suppose also that we randomly select one of these coins and then continually flip it. Let Hj denote the event that flip j, j ≥ 1, lands heads. Also, let Ci be the event that coin i
In a game series played with two teams, the first team to win a total of 4 games is the winner. Suppose that each game played is independently won by team A with probability p.a. Given that one team leads 3 to 0, what is the probability that it is team A that leads.b. Given that one team leads 3 to
If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events? Explain your reasoning.a. E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes.b. E is the
Suppose that you continually collect coupons and that there are m different types. Suppose also that each time a new coupon is obtained, it is a type i coupon with probability pi, i = 1, ..., m. Suppose that you have just collected your nth coupon. What is the probability that it is a new type?
An engineering system consisting of components is said to be a -out-of-system (k ≤ n) if the system functions if and only if at least k of the n components function. Suppose that all components function independently of one another.a. If the ith component functions with probability pi, i = 1, 2,
An investor owns shares in a stock whose present value is 25. She has decided that she must sell her stock if it goes either down to 10 or up to 40. If each change of price is either up 1 point with probability .55 or down 1 point with probability .45, and the successive changes are independent,
A and B flip coins. starts and continues flipping until a tail occurs, at which point starts flipping and continues until there is a tail. Then takes over, and so on. Let P1 be the probability of the coin landing on heads when flips and P2 when flips. The winner of the game is the first one to
An urn contains 12 balls, of which 4 are white. Three players—A, B, and successively C draw from the urn, A first, then B, then C, then A, and so on. The winner is the first one to draw a white ball. Find the probability of winning for each player ifa. Each ball is replaced after it is
Consider an eight team tournament with the format given in Figure 3.6. If the probability that team beats team if they play is find the probability that team wins the tournament. (1,8) (4,5) (3,6) (2,7)
Each of workers is independently qualified to do an incoming job with probability p. If none of them is qualified then the job is rejected; otherwise the job is assigned to a randomly chosen one of the qualified workers. Find the probability that worker is assigned to the first incoming job.
Suppose in the preceding problem that and that worker is qualified with probabilitya. Find the probability that worker is assigned to the first incoming job.b. Given that worker is assigned to the first job, find the conditional probability that worker was qualified for that job.
If has distribution function F, what is the distribution function of ex?
Three dice are rolled. By assuming that each of the 63 = 216 possible outcomes is equally likely, find the probabilities attached to the possible values that X can take on, where is the sum of the 3 dice.
The random variable is said to have the Yule-Simons distribution ifa. Show that the preceding is actually a probability mass function. That is, show thatb. Show that E[X] = 2.c. Show that E[X2] = ∞. 4 P{X = n} = , п21 п(п + 1)(п + 2)
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed times. What are the possible values of X?
Let be such thatP{X = 1} = p = 1 - P{X = - 1}Find c ≠ 1 such that E[cX] = 1.
Suppose that a die is rolled twice. What are the possible values that the following random variables can take on:a. The maximum value to appear in the two rolls;b. The minimum value to appear in the two rolls;c. The sum of the two rolls;d. The value of the first roll minus the value of the second
Let X be the winnings of a gambler. Let p(i) = P(X = i) and suppose thatCompute the conditional probability that the gambler wins i , i = 1, 2, 3, given that he wins a positive amount. 1/3: p(1) %3 р(— 1) %3D 13/55%; Р(2) %3D р(- 2)%3D 1/11:p(3) %3 р(-3) %3D 1/165 p(0)
The random variable is said to follow the distribution of Benford’s Law ifIt has been shown to be a good fit for the distribution of the first digit of many real life data values.a. Verify that the preceding is a probability mass function by showing thatb. Find P(X ≤ j). i+1 P(X = i) = log,10 i
In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess
A family has children with probability qpn, n ≥ 1, where α ≤ (1 - p)/p.a. What proportion of families has no children?b. If each child is equally likely to be a boy or a girl (independently of each other), what proportion of families consists of boys (and any number of girls)?
A deck of n cards numbered through are to be turned over one a time. Before each card is shown you are to guess which card it will be. After making your guess, you are told whether or not your guess is correct but not which card was turned over. It turns out that the strategy that maximizes the
Suppose that the distribution function of is given bya. Find P{X = i}, i = 1, 2, 3.b. Find P{1/2 < X < 3/2}. b< 0 b 0sb
Let X be a Poisson random variable with parameter λ. Show that P{X = i} increases monotonically and then decreases monotonically as i increases, reaching its maximum when is the largest integer not exceeding λ.
Four independent flips of a fair coin are made. Let X denote the number of heads obtained. Plot the probability mass function of the random variable X - 2.
Let X be a Poisson random variable with parameter λ.a. Show that by using the result of Theoretical Exercise 4.15 and the relationship between Poisson and binomial random variables.b. Verify the formula in part (a) directly by making use of the expansion of e-λ + eλ.A family has children
If the distribution function of X is given bycalculate the probability mass function of X. 0. b
One of the numbers 1 through 10 is randomly chosen. You are to try to guess the number chosen by asking questions with “yes–no” answers. Compute the expected number of questions you will need to ask in each of the following two cases:a. Your ith question is to be “Is it ?” i = 1, 2, 3, 4,
Suppose that the number of events that occur in a specified me is a Poisson random variable with parameter λ. If each event is counted with probability p, independently of every other event, show that the number of events that are counted is a Poisson random variable with parameter λp. Also, give
An insurance company writes a policy to the effect that an amount of money must be paid if some event occurs within a year. If the company estimates that will E occur within a year with probability p, what should it charge the customer in order that its expected profit will be 10 percent of A?
A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.
A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with n = 10, p = 1/3, approximately how many papers should he purchase so as to maximize his expected profit?
A jar contains n chips. Suppose that a boy successively draws a chip from the jar, each time replacing the one drawn before drawing another. The process continues until the boy draws a chip that he has previously drawn. Let X denote the number of draws, and compute its probability mass function.
In Example 4b, suppose that the department store incurs an additional cost of c for each unit of unmet demand. (This type of cost is often referred to as a goodwill cost because the store loses the goodwill of those customers whose demands it cannot meet.) Compute the expected profit when the store
If the die in Problem 4.7 is assumed fair, calculate the probabilities associated with the random variables in parts (a) through (d).Data from Problem 4.7a. The maximum value to appear in the two rolls;b. The minimum value to appear in the two rolls;c. The sum of the two rolls;d. The value of the
If balls are randomly chosen from an urn containing red, white, blue, and green balls, find the probability thata. At least one of the balls chosen is green;b. One ball of each color is chosen.
Consider two independent tosses of a fair coin. Let A be the event that the first toss results in heads, let B be the event that the second toss results in heads, and let be the event A, B, that in both tosses the coin lands on the same side. Show that the events A, B, and C are pairwise
Ninety-eight percent of all babies survive delivery. However, 15 percent of all births involve Cesarean (C) sections, and when a C section is performed, the baby survives 96 percent of the time. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby
Consider two urns, each containing both white and black balls. The probabilities of drawing white balls from the first and second urns are, respectively, p and p' Balls are sequentially selected with replacement as follows: With probability α, a ball is initially chosen from the first urn, and
In how many ways can 8 people be seated in a row ifa. There are no restrictions on the seating arrangement?b. Persons and must sit next to each other?c. There are 4 men and 4 women and no 2 men or 2 women can sit next to each other?d. There are 5 men and they must sit next to one another?e. There
Prove the following relations: Ů E; F = & E,F and O E;F and 1 1 (R E.) UF = n (E, U F). (E; U F). 1

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