- Suppose X1, . . . , Xn are independent exponential(λ). Show that min{X1, . . . , Xn} = exponential (nλ)
- Suppose X and Y have joint density f (x, y) = 6y when x > 0, y > 0, and x + y < 1.(a) Find the marginal densities of X and Y and(b) The conditional density of X given Y = y.
- At Llenroc College, 63% of freshmen who are premed switch to a liberal arts major, while 18% of liberal arts majors switch to being premed. If the incoming freshman class is 60% premed and 40%
- A person is flipping a coin repeatedly. Let Xn be the outcome of the two previous coin flips at time n; for example, the state might be HT to indicate that the last flip was T and the one before that
- A taxicab driver moves between the airport A and two hotels Band C according to the following rules. If he is at the airport, he will go to one of the two hotels next with equal probability. If at a
- Consider a gambler’s ruin chain with N = 4. That is, if 1 ≤ i ≤ 3, p(i, i + 1) = 0.4, and p(i, i − 1) = 0.6, but the endpoints are absorbing states: p(0, 0) = 1 and p(4, 4) = 1. Compute
- An outdoor restaurant in a resort town closes when it rains. From past records it was found that from May to September, when it rains one day the probability that it rains the next is 0.4; when it
- Market research suggests that in a 5-year period 8% of people with cable television will get rid of it and 26% of those without it will sign up for it. Compare the predictions of the Markov chain
- A sociology professor postulates that in each decade 8% of women in the workforce leave it and 20% of the women not in it begin to work. Compare the predictions of his model with the following data
- The following transition probability describes the migration patterns of birds between three habitats:If there are 1,000 birds in each habitat at the beginning of the first year, how many do we
- A car rental company has rental offices at both Kennedy and LaGuardia airports. Assume that a car rented at one airport must be returned to one of the two airports. If the car was rented at LaGuardia
- The 1990 census showed that 36% of the households in the District of Columbia were homeowners, while the remainder were renters. During the next decade 6% of the homeowners became renters and 12% of
- Most railroad cars are owned by individual railroad companies. When a car leaves its home railroad’s trackage, it becomes part of the national pool of cars and can be used by other railroads. A
- A rapid transit system has just started operating. In the first month of operation, it was found that 25% of commuters are using the system, while 75% are traveling by automobile. Suppose that each
- A regional health study indicates that from one year to the next, 75% of smokers will continue to smoke, while 25% will quit. 8% of those who stopped smoking will resume smoking, while 92% will not.
- The town of Mythica has a “free bikes for the people program.” You can pick up bikes at the library (L), the coffee shop (C), or the cooperative grocery store (G). The director of the program has
- Bob eats lunch at the campus food court every week day. He either eats Chinese food, Quesadilla, or Salad. His transition matrix isHe had Chinese food on Monday. What are the probabilities for his
- Census results reveal that in the United States 80% of the daughters of working women work and that 30% of the daughters of nonworking women work. (a) Write the transition probability for this model.
- Three of every four trucks on the road are followed by a car, while only one of every five cars is followed by a truck. What fraction of vehicles on the road are trucks?
- In a test paper the questions are arranged so that 3/4’s of the time a true answer is followed by a true, while 2/3’s of the time a false answer is followed by a false. You are confronted with a
- When a basketball player makes a shot then he tries a harder shot the next time and hits (H)with probability 0.4 and misses (M)with probability 0.6.When he misses he is more conservative the next
- A regional health study shows that from1 year to the next 76% of the people who smoked will continue to smoke and 24% will quite. 8% of those who do not smoke will start smoking, while 92% of those
- In unprofitable times corporations sometimes suspend dividend payments. Suppose that after a dividend has been paid the next one will be paid with probability 0.9,while after a dividend is suspended
- A university computer room has 30 terminals. Each day there is a 3% chance that a given terminal will break and a 72% chance that a given broken terminal will be repaired. Assume that the fates of
- A plant species has red, pink, or white flowers according to the genotypes RR, RW, and WW, respectively. If each of these genotypes is crossed with a pink (RW) plant then the offspring fractions
- A certain town never has two sunny days in a row. Each day is classified as rainy, cloudy, or sunny. If it is sunny one day then it is equally likely to be cloudy or rainy the next. If it is cloudy
- Amidwestern university has three types of health plans: a health maintenance organization (HMO), a preferred provider organization (PPO), and a traditional fee for service plan (FFS). In 2000, the
- A sociologist studying living patterns in a certain region determines that the pattern of movement between urban (U), suburban (S), and rural areas (R) is given by the following transition matrix:In
- In a large metropolitan area, commuters either drive alone (A), carpool (C), or take public transportation (T). A study showed that 80% of those who drive alone will continue to do so next year,
- In a particular county voters declare themselves as members of the Republican, Democrat, or Green party. No voters change directly from the Republican to Green party or vice versa. In a given year
- (a) Three telephone companies A, B, and C compete for customers. Each year A loses 5% of its customers to B and 20% to C; B loses 15% of its customers to A and 20% to C; C loses 5% of its customers
- An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. No one moves from poor to preferred or from preferred to poor in 1 year. 40% of the
- A professor has two lightbulbs in his garage. When both are burned out, they are replaced, and the next day starts with two working lightbulbs. Suppose that when both are working, one of the two will
- An individual has three umbrellas; some at her office and some at home. If she is leaving home in the morning (or leaving work at night) and it is raining, she will take an umbrella, if one is there.
- At the end of a month, a large retail store classifies each of its customer’s accounts according to current (0), 30–60 days overdue (1), 60–90 days overdue (2), more than 90 days overdue (3).
- At the beginning of each day, a piece of equipment is inspected to determine its working condition, which is classified as state 1 = new, 2, 3, or 4 = broken. We assume the state is a Markov chain
- To make a crude model of a forest we might introduce states 0 = grass, 1 = bushes, 2 = small trees, 3 = large trees, and write down a transition matrix such as the following:The idea behind this
- Five white balls and five black balls are distributed in two urns in such a way that each urn contains five balls. At each step we draw one ball from each urn and exchange them. Let Xn be the number
- Two competing companies are trying to buy up all the farms in a certain area to build houses. In each year 10% of farmers sell to company 1, 20% sell to company 2, and 70% keep farming. Neither
- A warehouse has a capacity to hold four items. If the warehouse is neither full nor empty, the number of items in the warehouse changes whenever a new item is produced or an item is sold. Suppose
- The Macro soft Company gives each of its employees the title of programmer (P) or project manager (M). In any given year 70% of programmers remain in that position, 20% are promoted to project
- At a nationwide travel agency, newly hired employees are classified as beginners (B). Every 6 months the performance of each agent is reviewed. Past records indicate that transitions through the
- At a manufacturing plant, employees are classified as trainee (R), technician (T), or supervisor (S). Writing Q for an employee who quits we model their progress through the ranks as a Markov chain
- The two previous problems have the following form:Show that(a) The probability of being absorbed in A rather than B is ac/(a + b)(c + d) and(b) The expected time to absorption starting from 1 is
- Suppose three runners from team A and three runners from team B have a race. If all six runners have equal ability, what is the probability that the three runners from team A will finish first,
- Four men and four women are shipwrecked on a tropical island. How many ways can they(a) Form four male–female couples,(b) Get married if we keep track of the order in which the weddings
- Two balls are drawn from an urn with balls numbered from1 up to 10.What is the probability that the two numbers will differ by more (>) than 3?
- How can 5 black and 5 white balls be put into two urns to maximize the probability that a white ball is drawn when we draw from a randomly chosen urn?
- Suppose we draw k cards out of a deck. What is the probability that we do not draw an ace? Is the answer larger or smaller than (3/4)k?
- You and a friend each roll two dice. What is the probability that you both have the same two numbers?
- In a dice game the “dealer” rolls two dice, the player rolls two dice, and the player wins if his total is larger than the dealer’s. What is the probability that the player wins?
- What is the most likely total for the sum of four dice and what is its probability?
- Charlie draws 5 cards out of a deck of 52. If he gets at least three of one suit, he discards the cards not of that suit and then draws until he again has 5 cards. For example, if he gets 3 hearts, 1
- Suppose 60% of the people in a town will get exposed to flu in the next month. If you are exposed and not inoculated then the probability of your getting the flu is 80%, but if you are inoculated
- John takes the bus with probability 0.3 and the subway with probability 0.7. He is late 40% of the time when he takes the bus, but only 20% of the time when he takes the subway. What is the
- The population of Cyprus is 70% Greek and 30% Turkish. 20% of the Greeks and 10% of the Turks speak English. What fraction of the people of Cyprus speak English?
- You are going to meet a friend at the airport. Your experience tells you that the plane is late 70% of the time when it rains, but is late only 20% of the time when it does not rain. The weather
- Two boys have identical piggy banks. The older boy has 18 quarters and 12 dimes in his; the younger boy, 2 quarters and 8 dimes. One day the two banks get mixed up. You pick up a bank at random and
- Suppose that the number of children in a family has the following distribution:Assume that each child is independently a girl or a boy with probability 1/2 each.If a family is picked at random what
- A student is taking a multiple-choice test in which each question has four possible answers. She knows the answers to 50% of the questions, can narrow the choices down to two 30% of the time, and
- Three boys take turns shooting a basketball and have probabilities 0.2, 0.3,and 0.5 of scoring a basket. Compute the probabilities for each boy to get the first basket.
- Change the second and third probabilities in the last problem so that each boy has an equal chance of winning.
- The alpha fetal protein test is meant to detect spina bifida in unborn babies, a condition that affects 1 out of 1,000 children who are born. The literature on the test indicates that 5% of the time
- Binary digits, that is, 0’s and 1’s, are sent down a noisy communications channel. They are received as sent with probability 0.9 but errors occur with probability 0.1.Assuming that 0’s and
- Two hunters shoot at a deer, which is hit by exactly one bullet. If the first hunter hits his target with probability 0.3 and the second with probability 0.6, what is the probability that the second
- A cab was involved in a hit-and-run accident at night. Two cab companies green and blue operate 85% and 15% of the cabs in the city respectively. A witness identified the cab as blue. However, in a
- A student goes to class on a snowy day with probability 0.4, but on a non snowy day attends with probability 0.7. Suppose that 20% of the days in February are snowy. What is the probability that it
- A company gave a test to 100 salesman, 80 with good sales records and 20 with poor sales records. 60% of the good salesman passed the test but only 30% of the poor salesmen did. Andy passed the test.
- A company rates 80% of its employees as satisfactory and 20% as unsatisfactory. Personnel records indicate that 70% of the satisfactory workers had prior experience but only 40% of the unsatisfactory
- A golfer hits his drive in the fairway with probability 0.7. When he hits his drive in the fairway he makes par 80% of the time. When he doesn’t, he makes par only 30% of the time. He just made par
- You are about to have an interview for Harvard Law School. 60% of the interviewers are conservative and 40% are liberal. 50% of the conservatives smoke cigars but only 25% of the liberals do. Your
- Five pennies are sitting on a table. One is a trick coin that has heads on both sides, but the other four are normal. You pick up a penny at random and flip it four times, getting heads each time.
- One slot machine pays off 1/2 of the time, while another pays off 1/4 of the time. We pick one of the machines and play it 6 times, winning 3 times. What is the probability we are playing the machine
- A student is taking a multiple-choice exam in which each question has four possible answers. She knows the answers to 60% of the questions and guesses at the others. What is the probability that she
- 20% of people are “accident-prone” and have a probability 0.15 of having an accident in a 1-year period in contrast to a probability of 0.05 for the other 80% of people.(a) If we pick a person
- One die has 4 red and 2 white sides; a second has 2 red and 4 white sides.(a) If we pick a die at random and roll it, what is the probability that the result is a red side?(b) If the first result is
- A particular football team is known to run 40% of its plays to the left and 60% to the right. When the play goes to the right, the right tackle shifts his stance 80% of the time, but does so only 10%
- A company gives a test to 100 salesmen, 80 with good sales records and 20 with poor records. 60% of the good salesmen pass the test, but only 30% of the poor salesmen do. A new applicant takes the
- You are a serious student who studies on Friday nights but your roommate goes out and has a good time. 40% of the time he goes out with his girlfriend; 60% of the time he goes to a bar. 30% of the
- Two masked robbers try to rob a crowded bank during the lunch hour but the teller presses a button that sets off an alarm and locks the front door. The robbers, realizing they are trapped, throw away
- Three bags lie on the table. One has two gold coins, one has two silver coins, and one has one silver and one gold. You pick a bag at random, and pick out one coin. If this coin is gold, what is the
- In a certain city, 30% of the people are Conservatives, 50% are Liberals, and 20% are Independents. In a given election, 2/3 of the Conservatives voted, 80% of the Liberals voted, and 50% of the
- An undergraduate student has asked a professor for a letter of recommendation. He estimates that the probability he will get the job is 0.8 with a strong letter, 0.4 with a medium letter, and 0.1
- A group of 20 people go out to dinner. 10 go to an Italian restaurant, 6 to a Japanese restaurant, and 4 to a French restaurant. The fractions of people satisfied with their meals were 4/5, 2/3, and
- 1 out of 1,000 births results in fraternal twins; 1 out of 1,500 births results in identical twins. Identical twins must be the same sex but the sexes of fraternal twins are independent. If two girls
- Consider the following data on traffic accidents:Calculate.(a) The probability that a randomly chosen driver will have an accident this year,(b) The probability that a driver is between 46 and 65
- Suppose we draw two tickets from a hat that contains tickets numbered 1, 2, 3, 4. Let X be the first number drawn and Y be the second. Find the joint distribution of X and Y.
- Suppose we roll one die repeatedly and let Ni be the number of the roll on which i first appears. Find the joint distribution of N1 and N6.
- Compute(a) P (X = 1|Y = 1)(b) P (X = 2|Y = 2)for the following joint distribution: Y 1 2 3 X = 1 0.1 0.15 0.05 2 0.2 0.15 0 3 0.3 0 0.05
- Compute(a) P (X = 2|Y = 3)(b) P (Y = 3|X = 3)for the following joint distribution: Y 1 2 3 X = 1 0.2 0.10 0.05 2 0.15 0 0.15 3 0.05 0.10 0.20
- Using the clues given below, fill in the rest of the joint distribution. There is only one answer. Y 3 1 ? ? 2 0.1 0.05 ? (a) P(Y=2|X = 0) = 1/4, (b) X and Y are independent. X = 0 6
- Using the clues given below, fill in the rest of the joint distribution. There is only one answer: Y 1 X = 1 ? 2 3 ? 0 ? ? 0 2 3 0 For k= 1, 2, 3, (a) P(Y = 1|X = k) = 2/3, (b) P(X = k|Y = 1) = k/6.
- Fill in the rest of the joint distribution so that X and Y are independent. There are two possible answers: Y X = 0 1 0 ? 2/9 2/9 ? 1 2.
- Find the mean and variance of the number of games in the World Series.Recall that it is won by the first team to win four games and assume that the outcomes are determined by flipping a coin.
- In a group of five items, two are defective. Find the distribution of N, the number of draws we need to find the first defective item. Find the mean and variance of N.
- A die is rolled 8 times. What is the probability that we will get exactly two 3’s?
- A random variable has P (X = x) = x/15 for x = 1, 2, 3, 4, 5, and 0 otherwise.Find the mean and variance of X.
- Suppose X has density function f (x) = c(3 − |x|) when −3 < x < 3.What value of c makes this a density function?
- Consider f (x) = c (1 − x2) for −1 < x < 1 and 0 otherwise. What value of c should we take to make f a density function?
- Suppose X has density function 6x(1 − x) for 0 < x < 1 and 0 otherwise.Find(a) E X,(b) E (X2),(c) var (X).

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