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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

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
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
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
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 =
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
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
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
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
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
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
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
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)
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
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
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
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,
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
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
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
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
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
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,
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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 ∈
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
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
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 =
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,
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
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
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,
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
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,
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
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
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
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
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
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,
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
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 =
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
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
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
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
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 =
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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) −

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