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essentials of stochastic processes

Essentials Of Stochastic Processes 3rd Edition Richard Durrett - Solutions

Queen’s random walk. A queen can move any number of squares horizontally, vertically, or diagonally. Let Xn be the sequence of squares that results if we pick one of queen’s legal moves at random. Find (a) the stationary distribution and(b) the expected number of moves to return to corner (1,1)
King’s random walk. This and the next example continue Example 1.30. A king can move one squares horizontally, vertically, or diagonally. Let Xn be the sequence of squares that results if we pick one of king’s legal moves at random.Find (a) the stationary distribution and (b) the expected
Random walk on a clock. Consider the numbers 1; 2; : : : 12 written around a ring as they usually are on a clock. Consider a Markov chain that at any point jumps with equal probability to the two adjacent numbers. (a)What is the expected number of steps that Xn will take to return to its starting
Library chain. On each request the ith of n possible books is the one chosen with probability pi. To make it quicker to find the book the next time, the librarian moves the book to the left end of the shelf. Define the state at any time to be the sequence of books we see as we examine the shelf
Bernoulli–Laplace model of diffusion. Consider two urns each of which contains m balls; b of these 2m balls are black, and the remaining 2m b are white.We say that the system is in state i if the first urn contains i black balls and m i white balls while the second contains b i black balls and
Consider a general chain with state space S D f1; 2g and write the transition probability asUse the Markov property to show thatand then concludeThis shows that if 0 1 2 11-a a 2 b 1-b
3. Two barbers and two chairs. Consider the following chain(a) Find the stationary distribution. (b) Compute Px.V0 (c) Let D minfV0; V4g. Find Ex for x D 1; 2; 3. 0 1 234 0 0 001 10 10.6 0 0.4 0 0 20 0.75 0 0.25 0 3 0 0 0.75 0 0.25 4 0 0 0 0.75 0.25
Customers shift between variable rate loans (V), thirty year fixed-rate loans(30), fifteen year fixed-rate loans (15), or enter the states paid in full (P), or foreclosed according to the following transition matrix:(a) For each of the three loan types find (a) the expected time until paid or
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 with transition probability R T S Q R :2 :6 0 :2 T 0 :55 :15 :3 S 0 0 1 0 Q 0 0 0 1(a) What fraction of
At a nationwide travel agency, newly hired employees are classified as beginners (B). Every six months the performance of each agent is reviewed. Past records indicate that transitions through the ranks to intermediate (I) and qualified(Q) are according to the following Markov chain, where F
The Megasoft 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 manager and 10% are fired (state X). 95%of project managers remain in that position while 5% are fired. How
At the New York State Fair in Syracuse, Larry encounters a carnival game where for one dollar he may buy a single coupon allowing him to play a guessing game. On each play, Larry has an even chance of winning or losing a coupon. When he runs out of coupons he loses the game. However, if he can
Sucker bet. Consider the following gambling game. Player 1 picks a three coin pattern (for example, HTH) and player 2 picks another (say THH). A coin is flipped repeatedly and outcomes are recorded until one of the two patterns appears.Somewhat surprisingly player 2 has a considerable advantage in
To find the waiting time for HTH we let the state of our Markov chains be the part of the pattern we have so far. The transition probability is 0 H HT HTH 0 0:5 0:5 0 0 H 0 0:5 0:5 0 HT 0:5 0 0 0:5 HTH 0 0 0 1(a) Find E0THTH. (b) use the reasoning for part (b) of the previous exercise to conclude
To find the waiting time for HHH we let the state of our Markov chains be the number of consecutive heads we have at the moment. The transition probability is 0 1 2 3 0 0:5 0:5 0 0 1 0:5 0 0:5 0 2 0:5 0 0 0:5 3 0 0 0 1 Find E0T3.(b) (10 points) Consider now the chain where the state gives the
Six children (Dick, Helen, Joni,Mark, Sam, and Tony) play catch. If Dick has the ball, he is equally likely to throw it to Helen, Mark, Sam, and Tony. If Helen has the ball, she is equally likely to throw it to Dick, Joni, Sam, and Tony. If Sam has the ball, he is equally likely to throw it to
The Duke football team can Pass, Run, throw an Interception, or Fumble.Suppose the sequence of outcomes is Markov chain with the following transition matrix.P R I F P 0:7 0:2 0:1 0 R 0:35 0:6 0 0:05 I 0 0 1 0 F 0 0 0 1 The first play is a pass. (a) What is the expected number of plays until a
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 that (no matter when we look) the probability that the next event is “a new item is produced” is
A bank classifies loans as paid in full (F), in good standing (G), in arrears (A), or as a bad debt (B). Loans move between the categories according to the following transition probability:F G A B F 1 0 0 0 G :1 :8 :1 0 A :1 :4 :4 :1 B 0 0 0 1 What fraction of loans in good standing are eventually
TheMarkov chain associated with a manufacturing process may be described as follows: A part to be manufactured will begin the process by entering step 1. After step 1, 20% of the parts must be reworked, i.e., returned to step 1, 10% of the parts are thrown away, and 70% proceed to step 2. After
Landscape dynamics. 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 like the following:0 1 2 3 0 1=2 1=2 0 0 1 1=24 7=8 1=12 0 2 1=36 0 8=9 1=12 3 1=8 0 0 7=8 The idea behind this matrix is that
At the beginning of each day, a piece of equipment is inspected to determine its working condition, which is classified as state 1 D new, 2, 3, or 4 D broken.Suppose that a broken machine requires three days to fix. To incorporate this into the Markov chain we add states 5 and 6 and write the
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 (3). Their experience indicates that the accounts move from state to state according to a Markov chain with
Reflecting random walk on the line. Consider the points 1; 2; 3; 4 to be marked on a straight line. Let Xn be a Markov chain that moves to the right with probability 2=3 and to the left with probability 1=3, but subject this time to the rule that if Xn tries to go to the left from 1 or to the right
Let Xn be the number of days since David last shaved, calculated at 7:30 a.m.when he is trying to decide if he wants to shave today. Suppose that Xn is a Markov chain with transition matrix 1 2 3 4 1 1=2 1=2 0 0 2 2=3 0 1=3 0 3 3=4 0 0 1=4 4 1 0 0 0 In words, if he last shaved k days ago, he 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. Otherwise, she gets wet. Assume that independent of the past, it rains on each trip with
A professor has two light bulbs in his garage. When both are burned out, they are replaced, and the next day starts with two working light bulbs. Suppose that when both are working, one of the two will go out with probability .02 (each has probability .01 and we ignore the possibility of losing two
An auto insurance company classifies its customers into three categories:poor, satisfactory, and excellent. No one moves from poor to excellent or from excellent to poor in one year.P S E P :6 :4 0 S :1 :6 :3 E 0 :2 :8 What is the limiting fraction of drivers in each of these categories?
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. Other transitions occur according to the following matrix:R D G R :85 :15 0 D :05 :85 :10 G 0 :05 :95 In the long run
(a) Three telephone companies A, B, and C compete for customers. Each year customers switch between companies according to the following transition probability A B C A :75 :05 :20 B :15 :65 :20 C :05 :1 :85 What is the limiting market share for each of these companies?
In a large metropolitan area, commuters either drive alone (A), carpool (C), or take public transportation (T). A study showed that transportation changes according to the following matrix:A C T A :8 :15 :05 C :05 :9 :05 S :05 :1 :85 In the long run what fraction of commuters will use the three
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.U S R U :86 :08 :06 S :05 :88 :07 R :03 :05 :92 In the long run what fraction of the population will
The weather in a certain town is classified as rainy, cloudy, or sunny and changes according to the following transition probability is R C S R 1=2 1=4 1=4 C 1=4 1=2 1=4 S 1=2 1=2 0 In the long run what proportion of days in this town are rainy? cloudy? sunny?
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 are RR RW WW RR :5 :5 0 RW :25 :5 :25 WW 0 :5 :5 What is the long run fraction of plants of the three
The liberal town of Ithaca 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 determined that bikes move around according to the following Markov chain L C G L :5 :2 :3 C
Bob eats lunch at the campus food court every week day. He either eats Chinese food, Quesadila, or Salad. His transition matrix isHe had Chinese food on Monday. (a) What are the probabilities for his three meal choices on Friday (four days later). (b) What are the long run frequencies for his three
A midwestern 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). Experience dictates that people change plans according to the following transition matrixIn 2000, the percentages
(a) Suppose brands A and B have consumer loyalties of .7 and .8, meaning that a customer who buys A one week will with probability .7 buy it again the next week, or try the other brand with .3. What is the limiting market share for each of these products? (b) Suppose now there is a third brand with
Folk wisdom holds that in Ithaca in the summer it rains 1/3 of the time, but a rainy day is followed by a second one with probability 1/2. Suppose that Ithaca weather is a Markov chain. What is its transition probability?
When a basketball player makes a shot then he tries a harder shot the next time and hits (H) with probability 0.4, misses (M) with probability 0.6. When he misses he is more conservative the next time and hits (H) with probability 0.7, misses (M)with probability 0.3. (a) Write the transition
Census results reveal that in the USA 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. (b) In the long run what fraction of women will be working?
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 the next one will be suspended with probability 0.6. In the long run what is the fraction of
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 100 question test paper. Approximately what fraction of the answers will be True.
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?
A regional health study indicates that from one year to the next, 75% percent of smokers will continue to smoke while 25% will quit. 8% of those who stopped smoking will resume smoking while 92% will not. If 70% of the population were smokers in 1995, what fraction will be smokers in 1998? in 2005?
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 travelling by automobile. Suppose that each month 10% of transit users go back to using their cars, while 30% of automobile users switch to the
A sociology professor postulates that in each decade 8% of women in the work force leave it and 20% of the women not in it begin to work. Compare the predictions of his model with the following data on the percentage of women working: 43.3% in 1970, 51.5% in 1980, 57.5% in 1990, and 59.8% in 2000.
Market research suggests that in a five 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 model with the following data on the fraction of people with cable TV: 56.4% in 1990, 63.4% in
Find limn!1 pn.i; j/ for the chains in parts (c), (d), and (e) of Problem 1.8.
Do the following Markov chains converge to equilibrium? (a) 1 2 3 4 100 10 200.5.5 3.3.700 4 1000 (c) 1 2 3 4 5 6 10.5.5 0 0 0 2000 100 3 0 0 0.4 0.6 4 1 0 0 0 0 0 5 0 1 0 0 0 0 6 .200 0.80 (b) 1 2 3 4 10 1.0 0 2000 1 3 1 0 0 0 4 1/302/30 (d) 1 2 3 4 5 6 1 0 0 1 0 0 0 2 1 0 0 0 0 0 3 0.50 0.5 0 4
Consider the Markov chain with transition matrix:(a) Compute p2. (b) Find the stationary distributions of p and all of the stationary distributions of p2. (c) Find the limit of p2n.x; x/ as n!1. 1 2 3 4 1 0 0 0.1 0.9 2 0 0 0.6 0.4 3 0.8 0.2 0 0 4 0.4 0.6 0 0
Find the stationary distributions for the chains in exercises (a) 1.2, (b) 1.3, and (c) 1.7.
(a) Find the stationary distribution for the transition probability(b) Does it satisfy the detailed balance condition (1.11)? 123 1.2.4.4 2.1.6.3 3.2.6.2
Find the stationary distributions for the Markov chains on f1; 2; 3; 4g with transition matrices: 7.7 0.3 0 .60.40 .7.30 01 .2.5.30 .7 0.3 0 .2.5.30 (a) (b) (c) 0.5 0.5 .0.3.6.1 .1.2.4.3 0.4 0.6 0 0.2.8 0.3 0.7,
Find the stationary distributions for the Markov chains with transitionmatrices: (a) 1 2 3 1.5.4.1 (b) 123 1.5.4.1 (c) 123 1.6.4 0 2.2.5.3 2.3.4.3 2.2.4.2 3.1.3.6 3.2.2.6 30.2.8
Consider the following transition matrices. Identify the transient and recurrent states, and the irreducible closed sets in the Markov chains. Give reasons for your answers. (a) 1 2 3 4 5 1.4.3.3 0 0 2 0.5 0.5 0 3.50.500 4 0.5 0.50 5 0.3 0.3 .4 (c) 1 2 3 4 5 1 0 0 0 0 1 2 0.2 0.80 3.1.2.4.3 0 4 0.4
Suppose that the probability it rains today is 0.3 if neither of the last two days was rainy, but 0.6 if at least one of the last two days was rainy. Let the weather on day n, Wn, be R for rain, or S for sun. Wn is not a Markov chain, but the weather for the last two days Xn D .Wn1;Wn/ is a Markov
A taxicab driver moves between the airport A and two hotels B and C according to the following rules. If he is at the airport, he will be at one of the two hotels next with equal probability. If at a hotel, then he returns to the airport with probability 3/4 and goes to the other hotel with
Consider a gambler’s ruin chain with N D 4. That is, if 1 i 3, p.i; iC1/ D 0:4, and p.i; i 1/ D 0:6, but the endpoints are absorbing states: p.0; 0/ D 1 and p.4; 4/ D 1 Compute p3.1; 4/ and p3.1; 0/.
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 the renters became homeowners. What percentage were homeowners in 2000? in 2010?
We repeated roll two four sided dice with numbers 1, 2, 3, and 4 on them. Let Yk be the sum on the kth roll, Sn D Y1 C CYn be the total of the first n rolls, and Xn D Sn .mod 6/. Find the transition probability for Xn.
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 of white balls in the left urn at time n. Compute the transition probability for Xn.
A fair coin is tossed repeatedly with results Y0; Y1; Y2; : : : that are 0 or 1 with probability 1/2 each. For n 1 let Xn D Yn C Yn1 be the number of 1’s in the.n 1/th and nth tosses. Is Xn a Markov chain?
Bob, who has 15 pennies, and Charlie, who has 10 pennies, decide to play a game. They each flip a coin. If the two coins match, Bob gets the two pennies (for a profit of 1). If the two coins are different, then Charlie gets the two pennies. They quit when someone has all of the pennies. What is the
In tennis the winner of a game is the first player to win four points, unless the score is 4 3, in which case the game must continue until one player is ahead by two points and wins the game. Suppose that the server win the point with probability 0:6 and successive points are independent. What is
At a local two year college, 60% of freshmen become sophomores, 25% remain freshmen, and 15% drop out. 70% of sophomores graduate and transfer to a four year college, 20% remain sophomores and 10% drop out. What fraction of new students eventually graduate?
These processes arose from Francis Galton’s statistical investigation of the extinction of family names. Consider a population in which each individual in the nth generation independently gives birth, producing k children (who are members of generation n C 1) with probability pk. In Galton’s
Suppose there are three types of laundry detergent, 1, 2, and 3, and let Xn be the brand chosen on the nth purchase. Customers who try these brands are satisfied and choose the same thing again with probabilities 0.8, 0.6, and 0.4, respectively. When they change they pick one of the other two
Let Xn be a family’s social class in the nth generation, which we assume is either 1 = lower, 2 = middle, or 3 = upper. In our simple version of sociology, changes of status are a Markov chain with the following transition probability 1 2 3 1 :7 :2 :1 2 :3 :5 :2 3 :2 :4 :4 Q. Do the fractions of

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