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Fundamentals Of Probability With Stochastic Processes 3rd Edition Saeed Ghahramani - Solutions
8. Construct a transition probability matrix of a Markov chain with state space {1, 2, . . . , 8} in which {1, 2, 3, 4} is transient having period 4, {5} is aperiodic transient, and {6, 7, 8} is recurrent having period 3.
7. On a given vacation day, a sportsman either goes horseback riding (activity 1), or sailing (activity 2), or scuba diving (activity 3). For 1 ≤ i ≤ 3, let Xn = i, if the sportsman devotes vacation day n to activity i. Suppose that{Xn : n = 1, 2,...}is a Markov chain, and depending on which of
6. A fair die is tossed repeatedly. Let Xn be the number of 6’s obtained in the first n tosses. Show that {Xn : n = 1, 2,...} is a Markov chain. Then find its transition probability matrix, specify the classes and determine which are recurrent and which are transient.
5. The following is the transition probability matrix of a Markov chain with state space {1, 2, 3, 4, 5}. Specify the classes and determine which classes are transient and which are recurrent. P = 0 0 0 0 1 0 1/3 0 2/3 0 001/201/2 0 0 0 1 0 002/5 0 3/5
4. Let {Xn : n = 0, 1,...} be a Markov chain with state space {0, 1} and transition probability matrix P = ; 2/5 3/5 1/3 2/3 < Starting from 0, find the expected number of transitions until the first visit to 1.
3. Show that the following matrices are the transition probability matrix of the same Markov chain with elements of the state space labeled differently. P1 = 2/5 0 0 3/5 0 1/3 1/3 0 1/3 0 0 0 1/201/2 1/4 0 0 3/4 0 0 0 1/302/3 , P2 =
2. A Markov chain with transition probability matrix P = (pij ) is called regular, if for some positive integer n, pn ij > 0 for all i and j . Let {Xn : n = 0, 1,...} be a Markov chain with state space {0, 1} and transition probability matrix P = ; 0 1 1 0< Is {Xn : n = 0, 1,...} regular? Why or
1. Jobs arrive at a file server at a Poisson rate of 3 per minute. If 10 jobs arrived within 3 minutes, between 10:00 and 10:03, what is the probability that the last job arrived after 40 seconds past 10:02?
11. Let V (t) be the price of a stock, per share, at time t. Suppose that the stock’s current value, per share, is $95.00 with drift parameter −$2 per year and variance parameter 5.29. If $ V (t): t ≥ 0 % is a geometric Brownian motion, what is the probability that after 9 months the stock
10. Suppose that liquid in a cubic container is placed in a coordinate system. Suppose that at time 0, a particle is at (0, 0, 0), the origin. Let ! X(t), Y (t), Z(t)" be the coordinates of the particle after t units of time, and assume that X(t), Y (t), and Z(t) are independent Brownian motions,
9. (Reflected Brownian Motion) Suppose that liquid in a cubic container is placed in a coordinate system in such a way that the bottom of the container is placed on the xy-plane. Therefore, whenever a particle reaches the xy-plane, it cannot cross the bottom of the container. So it reverberates
8. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. For u > 0, t ≥ 0, find E 4 X(t)X(t + u)5 .
7. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. For u > 0, show that E 4 X(t + u) | X(t)5 = X(t). Therefore, for s > t, E 4 X(s) | X(t)5 = X(t).
6. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. As we know, for t1 and t2, t1 < t2, the random variables X(t1) and X(t2) are not independent. Find the distribution of X(t1) + X(t2).
5. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. For a fixed t > 0, let T be the smallest zero greater than t. Find the probability distribution function of T .
4. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. Let Tα be the time of hitting α first. Let Y ∼ N (0, σ2/α2). Show that, for α > 0, Tα and 1/Y 2 are identically distributed.
3. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. For ε > 0, show that limt→0 P *|X(t)| t > ε , = 1, whereas limt→∞ P *|X(t)| t > ε , = 0.
2. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. Show that, for all t > 0, |X(t)| and max 0≤s≤t X(s) are identically distributed.
1. Suppose that liquid in a container is placed in a coordinate system, and at time 0, a pollen particle suspended in the liquid is at (0, 0, 0), the origin. Let Z(t) be the z-coordinate of the position of the pollen after t minutes. Suppose that $ Z(t): t ≥ 0 % is a Brownian motion with variance
18. Let $ X(t): t ≥ 0 % be a birth and death process with birth rates $ λn %∞ n=0 and death rates $ µn %∞ n=1. Show that if.∞ k=1 µ1µ2 ···µk λ1λ2 ··· λk = ∞, then, with probability 1, eventually extinction will occur.
17. (Birth and Death with Disaster) Consider a population of a certain colonizing species. Suppose that each individual produces offspring at a Poisson rate of λ as long as it lives. Furthermore, suppose that the natural lifetime of an individual in the population is exponential with mean 1/µ
16. (Tandem or Sequential Queueing System) In a computer store, customers arrive at a cashier desk at a Poisson rate of λ to pay for the goods they want to purchase. If the cashier is busy, then they wait in line until their turn on a firstcome, first-served basis. The time it takes for the
15. (TheYule Process) A cosmic particle entering the earth’s atmosphere collides with air particles and transfers kinetic energy to them. These in turn collide with other particles transferring energy to them and so on. A shower of particles results. Suppose that the time that it takes for each
14. Let $ N (t): t ≥ 0 % be a Poisson process with rate λ. By Example 12.38, the process $ N (t): t ≥ 0 % is a continuous-time Markov chain. Hence it satisfies equations (12.12), the Chapman-Kolmogorov equations. Verify this fact by direct calculations.
13. Recall that an M/M/c queueing system is a GI/G/c system in which there are c servers, customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/µ. Suppose that ρ = λ/(cµ) < 1; hence the queueing system is stable. Find the long-run
12. Johnson Medical Associates has two physicians on call practicing independently. Each physician is available to answer patients’ calls for independent time periods that are exponentially distributed with mean 1/λ. Between those periods, the physician takes breaks for independent exponential
11. Consider a pure death process with µn = µ, n > 0. For i, j ≥ 0, find pij (t).
10. (Birth and Death with Immigration) Consider a population of a certain colonizing species. Suppose that each individual produces offspring at a Poisson rate λ as long as it lives. Moreover, suppose that new individuals immigrate into the population at a Poisson rate of γ . If the lifetime of
9. In Springfield, Massachusetts, people drive their cars to a state inspection center for annual safety and emission certification at a Poisson rate of λ. For n ≥ 1, if there are n cars at the center either being inspected or waiting to be inspected, the probability is 1−αn that an
8. There are m machines in a factory operating independently. The factory has k (k < m) repairpersons, and each repairperson repairs one machine at a time. Suppose that (i) each machine works for a time period that is exponentially distributed with mean 1/µ, then it breaks down; (ii) the time
7. Consider an M/M/1 queuing system in which customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/λ. We know that, in the long run, such a system will not be stable. For i ≥ 0, suppose that at a certain time there are i customers in the
6. In Example 12.41, is the continuous-time Markov chain $ X(t): t ≥ 0 % a birth and death process?
5. (Erlang’s Loss System) Each operator at the customer service department of an airline can serve only one call. There are c operators, and the incoming calls form a Poisson process with rate λ. The time it takes to serve a customer is exponential with mean 1/µ, independent of other customers
4. An M/M/∞ queueing system is similar to an M/M/1 system except that it has infinitely many servers. Therefore, all customers will be served upon arrival, and there will not be a queue. Examples of infinite-server systems are service facilities that provide self-service such as libraries.
3. Taxis arrive at the pick up area of a hotel at a Poisson rate of µ. Independently, passengers arrive at the same location at a Poisson rate of λ. If there are no passengers waiting to be put in service, the taxis wait in a queue until needed. Similarly, if there are no taxis available,
2. The director of the study abroad program at a college advises one, two, or three students at a time depending on how many students are waiting outside his office. The time for each advisement session, regardless of the number of participants, is exponential with mean 1/µ, independent of other
1. Let $ X(t): t ≥ 0 % be a continuous-time Markov chain with state space S. Show that for i, j ∈ S and t ≥ 0, p7 ij (t) = . k,=i qikpkj (t) − νipij (t). In other words, prove Kolmogorov’s backward equations.
33. For a simple random walk {Xn : n = 0, ±1, ±2,...}, discussed in Examples 12.12 and 12.22, show that P (Xn = j | X0 = i) = ; n n + j − i 2 < p(n+j−i)/2 (1 − p)(n−j+i)/2 if n + j − i is an even nonnegative integer for which n + j − i 2 ≤ n, and it is 0 otherwise.
32. In this exercise, we will outline a third technique for solving Example 3.31: We draw cards, one at a time, at random and successively from an ordinary deck of 52 cards with replacement. What is the probability that an ace appears before a face card? Hint: Consider a Markov chain {Xn : n = 1,
31. Consider the branching process of Example 12.24. In that process, before death an organism produces j (j ≥ 0) offspring with probability αj . Let X0 = 1, and let µ, the expected number of offspring of an organism, be greater than 1. Let p be the probability that extinction will occur. Show
30. Consider the gambler’s ruin problem (Example 3.14) in which two gamblers play the game of “heads or tails.” Each time a fair coin lands heads up, player A wins $1 from player B, and each time it lands tails up, player B wins $1 from A. Suppose that, initially, player A has a dollars and
29. Every Sunday, Bob calls Liz to see if she will play tennis with him on that day. If Liz has not played tennis with Bob since i Sundays ago, the probability that she will say yes to him is i/k, k ≥ 2, i = 1, 2, ... , k. Therefore, if, for example, Liz does not play tennis with Bob for k − 1
28. Let {Xn : n = 0, 1,...} be a Markov chain with state space S and probability transition matrix P = (pij ). Show that periodicity is a class property. That is, for i, j ∈ S, if i and j communicate with each other, then they have the same period.
27. Consider a Markov chain with state space S. Let i, j ∈ S. We say that state j is accessible from state i in n steps if there is a path i = i1, i2, i3, . . . , in = j with i1, i2, . . . , in ∈ S and pimim+1 > 0, 1 ≤ m ≤ n − 1. Show that if S is finite having K states, and j is
26. Show that if P and Q are two transition probability matrices with the same number of rows, and hence columns, then PQ is also a transition probability matrix. Note that this implies that if P is a transition probability matrix, then so is Pn for any positive integer n.
25. Recall that an M/M/1 queueing system is a GI/G/c system in which customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/µ. For an M/M/1 queueing system, each time that a customer arrives to the system or a customer departs from the system,
24. For a two-dimensional symmetric random walk, defined in Example 12.22, show that (0, 0) is recurrent and conclude that all states are recurrent.
23. Let {Xn : n = 0, 1,...} be a Markov chain with state space S. For i0, i1, . . . , in, j ∈ S, n ≥ 0, and m > 0, show that P (Xn+m = j | X0 = i0, X1 = i1, . . . , Xn = in) = P (Xn+m = j | Xn = in).
22. Carl and Stan play the game of “heads or tails,” in which each time a coin lands heads up, Carl wins $1 from Stan, and each time it lands tails up, Stan wins $1 from Carl. Suppose that, initially, Carl and Stan have the same amount of money and, as necessary, will be funded equally so that
21. Let {Xn : n = 0, 1,...} be a random walk with state space {0, 1, 2,...} and transition probability matrix P = 1 − p p 0 0 0 0 ... 1 − p 0 p 0 0 0 ... 0 1 − p 0 p 0 0 ... 0 0 1 − p 0 p 0 ... 0 0 0 1 − p 0 p . . . . . .
20. Alberto andAngela play backgammon regularly. The probability thatAlberto wins a game depends on whether he won or lost the previous game. It is p for Alberto to win a game if he lost the previous game, and p to lose a game if he won the previous one. (a) For n > 1, show that if Alberto wins the
19. A fair die is tossed repeatedly. We begin studying the outcomes after the first 6 occurs. Let the first 6 be called the zeroth outcome, let the first outcome after the first six, whatever it is, be called the first outcome, and so forth. For n ≥ 1, define Xn = i if the last 6 before the nth
18. Mr. Gorfin is a movie buff who watches movies regularly. His son has observed that whether Mr. Gorfin watches a drama or not depends on the previous two movies he has watched with the following probabilities: 7/8 if the last two movies he watched were both dramas, 1/2 if exactly one of them was
17. In Example 12.13, at the Writing Center of a college, pk > 0 is the probability that a new computer needs to be replaced after k semesters. For a computer in use at the end of the nth semester, let Xn be the number of additional semesters it remains functional. Then {Xn : n = 0, 1,...} is a
16. Seven identical balls are randomly distributed among two urns. Step 1 of a game begins by flipping a fair coin. If it lands heads up, urn I is selected; otherwise, urn II is selected. In step 2 of the game, a ball is removed randomly from the urn selected in step 1. Then the coin is flipped
15. Consider an Ehrenfest chain with 5 balls (see Example 12.15). Find the expected number of balls transferred between two consecutive times that an urn becomes empty.
14. For Example 12.10, where a mouse is moving inside the given maze, find the probability that the mouse is in cell i, 1 ≤ i ≤ 9, at a random time in the future.
13. Three players play a game in which they take turns and draw cards from an ordinary deck of 52 cards, successively, at random and with replacement. Player I draws cards until an ace is drawn. Then player II draws cards until a diamond is drawn. Next, player III draws cards until a face card is
12. An observer at a lake notices that when fish are caught, only 1 out of 9 trout is caught after another trout, with no other fish between, whereas 10 out of 11 nontrout are caught following nontrout, with no trout between. Assuming that all fish are equally likely to be caught, what fraction of
11. On a given vacation day, a sportsman goes horseback riding (activity 1), sailing (activity 2), or scuba diving (activity 3). Let Xn = 1 if he goes horseback riding . on day n, Xn = 2 if he goes sailing on day n, and Xn = 3 if he goes scuba diving on that day. Suppose that {Xn : n = 1,
10. The following is the transition probability matrix of a Markov chain with state space {1, 2, . . . , 7}. Starting from state 6, find the probability that the Markov chain will eventually be absorbed into state 4. P = 0.3 0.7 0 0 0 0 0 0.3 0.2 0.5 0 0 0 0
9. Construct a transition probability matrix of a Markov chain with state space {1, 2, . . . , 8} in which {1, 2, 3} is a transient class having period 3, {4} is an aperiodic transient class, and {5, 6, 7, 8} is a recurrent class having period 2.
8. A fair die is tossed repeatedly. The maximum of the first n outcomes is denoted by Xn. Is {Xn : n = 1, 2,...} a Markov chain? Why or why not? If it is a Markov chain, calculate its transition probability matrix, specify the classes, and determine which classes are recurrent and which are
7. The following is the transition probability matrix of a Markov chain with state space {0, 1, 2, 3, 4}. Specify the classes, and determine which classes are transient and which are recurrent. P = 2/5 0 0 3/5 0 1/3 1/3 0 1/3 0 0 0 1/201/2 1/4 0 0 3/4 0 0 0 1/302/3
6. Consider an Ehrenfest chain with 5 balls (see Example 12.15). If the probability mass function of X0, the initial number of balls in urn I, is given by P (X0 = i) = i 15, 0 ≤ i ≤ 5, find the probability that, after 6 transitions, urn I has 4 balls.
5. On a given day, Emmett drives to work (state 1), takes the train (state 2), or hails a taxi (state 3). Let Xn = 1 if he drives to work on day n, Xn = 2 if he takes the train on day n, andXn = 3 if he hails a taxi on that day. Suppose that{Xn : n = 1, 2,...} is a Markov chain, and depending on
4. Let {Xn : n = 0, 1,...} be a Markov chain with state space {0, 1, 2} and transition probability matrix P = 1/2 1/4 1/4 2/3 1/3 0 001 . Starting from 0, what is the probability that the process never enters 1?
3. Consider a circular random walk in which six points 1, 2, ... , 6 are placed, in a clockwise order, on a circle. Suppose that one-step transitions are possible only from a point to its adjacent points with equal probabilities. Starting from 1, (a) find the probability that in 4 transitions the
2. For a Markov chain {Xn : n = 0, 1,...} with state space {0, 1, 2,...} and transition probability matrix P = (pij ), let p be the probability mass function of X0; that is, p(i) = P (X0 = i), i = 0, 1, 2, . . . . Find the probability mass function of Xn.
1. In a community, there are N male and M female residents, N , M > 1000. Suppose that in a study, people are chosen at random and are asked questions concerning their opinion with regard to a specific issue. Let Xn = 1 if the nth person chosen is female, and Xn = 0 otherwise. Is {Xn : n = 1,
10. There are k types of shocks identified that occur, independently, to a system. For 1 ≤ i ≤ k, suppose that shocks of type i occur to the system at a Poisson rate of λi. Find the probability that the nth shock occurring to the system is of type i, 1 ≤ i ≤ n. Hint: For 1 ≤ i ≤ k, let
9. Customers arrive at a bank at a Poisson rate of λ. Let M(t) be the number of customers who enter the bank by time t only to make deposits to their accounts. Suppose that, independent of other customers, the probability is p that a customer enters the bank only to make a deposit. Show that $
8. Recall that an M/M/1 queueing system is a GI/G/1 system in which there is one server, customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/µ. For an M/M/1 queueing system, (a) show that the number of arrivals during a period in which a
7. Recall that an M/M/1 queueing system is a GI/G/1 system in which there is one server, customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/µ. For anM/M/1 system, each time a customer arrives at or a customer departs from the system, we say
6. Let $ N (t): t ≥ 0 % be a Poisson process. For k ≥ 1, let Sk be the time that the kth event occurs. Show that E 4 Sk | N (t) = n 5 = kt n + 1 .
5. A wire manufacturing company has inspectors to examine the wire for fractures as it comes out of a machine. The number of fractures is distributed in accordance with a Poisson process, having one fracture on the average for every 60 meters of wire. One day an inspector had to take an emergency
4. Suppose that a fisherman catches fish at a Poisson rate of 2 per hour. We know that yesterday he began fishing at 9:00 A.M., and by 1:00 P.M. he caught 6 fish. What is the probability that he caught the first fish before 10:00 A.M.?
3. When Linda walks from home to work, she has to cross the street at a certain point. Linda needs a gap of 15 seconds in the traffic to cross the street at that point. Suppose that the traffic flow is a Poisson process, and the mean time between two consecutive cars passing by Linda is 7 seconds.
2. The number of accidents at an intersection is a Poisson process $ N (t): t ≥ 0 % with rate 2.3 per week. Let Xi be the number of injuries in accident i. Suppose that{Xi}is a sequence of independent and identically distributed random variables with mean 1.2 and standard deviation 0.7.
1. For a Poisson process with parameter λ, show that, for all ε > 0, P *D D D N (t) t − λ D D D ≥ ε , → 0, as t → ∞. This shows that, for a large t, N (t)/t is a good estimate for λ.
20. An ordinary deck of 52 cards is divided randomly into 26 pairs. Using Chebyshev’s inequality, find an upper bound for the probability that, at most, 10 pairs consist of a black and a red card. Hint: For i = 1, 2, . . . , 26, let Xi = 1 if the ith red card is paired with a black card and Xi =
19. Show that for a nonnegative random variable X with mean µ, we have that ∀n, nP (X ≥ nµ) ≤ 1.
18. A fair die is rolled 20 times. What is the approximate probability that the sum of the outcomes is between 65 and 75?
14. A psychologist wants to estimate µ, the mean IQ of the students of a university. To do so, she takes a sample of size n of the students and measures their IQ’s. Then she finds the average of these numbers. If she believes that the IQ’s of these students are independent random variables
13. For a coin, p, the probability of heads is unknown. To estimate p, we flip the coin 5000 times and let pJbe the fraction of times it lands heads up. Show that the probability is at least 0.98 that pJestimates p within ±0.05.
12. In a clinical trial, the probability of success for a treatment is to be estimated. If the error of estimation is not allowed to exceed 0.01 with probability 0.94, how many patients should be chosen independently and at random for the treatment group?
11. Let X¯ denote the mean of a random sample of size 28 from a distribution with µ = 1 and σ2 = 4. Approximate P (0.95 < X
10. Let X and Y be independent Poisson random variables with parameters λ and µ, respectively. (a) Show that P (X + Y = n) = .n i=0 P (X = i)P (Y = n − i). (b) Use part (a) to prove that X+Y is a Poisson random variable with parameter λ + µ.
9. Find the moment-generating function of a random variableX withLaplace density function defined by f (x) = 1 2 e−|x| , −∞ < x < ∞
7. The moment-generating function of a random variable X is given by MX(t) = 1 (1 − t)2 , t < 1. Find the moments of X.
6. The moment-generating function of X is given by MX(t) = exp *et − 1 2 , . Find P (X > 0).
5. Let the moment-generating function of a random variable X be given by MX(t) = 1 t (et/2 − e−t/2 ) if t ,= 0 1 if t = 0. Find the distribution function of X.
4. For a random variable X, suppose that MX(t) = exp(2t 2 + t). Find E(X) and Var(X).
3. The moment-generating function of a random variable X is given by MX(t) = 1 6 et + 1 3 e2t + 1 2 e3t . Find the distribution function of X.
2. The moment-generating function of a random variable X is given by MX(t) = *1 3 + 2 3 et ,10 . Find Var(X) and P (X ≥ 8).
13. Let {X1, X2,...} be a sequence of independent Poisson random variables, each with parameter 1. By applying the central limit theorem to this sequence, prove that lim n→∞ 1 en .n k=0 nk k! = 1 2 .
12. Let{X1, X2,...} be a sequence of independent standard normal random variables. Let Sn = X2 1 + X2 2 + · · · + X2 n. Find lim n→∞ P (Sn ≤ n + √ 2n ). Hint: See Example 11.11.
11. A fair coin is tossed successively. Using the central limit theorem, find an approximation for the probability of obtaining at least 25 heads before 50 tails.
9. Consider a distribution with mean µ and probability density function f (x) = 1 x ln(3/2) 4 ≤ x ≤ 6 0 elsewhere. Determine the values of n for which the probability is at least 0.98 that the mean of a random sample of size n from the population is within ±0.07 of µ? B
5. Let X1, X2, . . . , Xn be independent and identically distributed random variables, and let Sn = X1 +X2 +· · · +Xn. For large n, what is the approximate probability that Sn is between E(Sn) − σSn and E(Sn) + σSn ?
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