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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

5. Consider a one-teller bank which operates as an M/M/1 queueing system with firstcome, first-served service order. Suppose that customers arrive at a Poisson rate of λ, and their service times are
4. 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
3. Consider anM/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
2. 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, and the time until the first individual arrives
1. An urn contains 7 red, 11 blue, and 13 yellow balls. Carmela, Daniela, and Lucrezia play a game in which they take turns and draw balls from the urn, successively, at random and with replacement.
15. (Death Process with Immigration) Consider a population of size n, n ≥ 0, of a certain species in which individuals do not reproduce. However, new individuals immigrate into the population at a
14. 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 ≥ 0, if there are n cars at the
13. There are m machines in a factory operating independently. Each machine works for a time period that is exponentially distributed with mean 1/μ. Then it breaks down.The time that it takes to
12. In a factory, there are m operating machines and s machines used as spares and ready to operate. The factory has k repair persons, and each repair person repairs one machine at a time. Suppose
11. Passengers arrive at a train station according to a Poisson process with rate λ and, independently, trains arrive at the same station according to another Poisson process, but with the same rate
10. On a given vacation day, Francesco either plays golf (activity 1) or tennis (activity 2).For i = 1, 2, letXn = i, if Francesco devotes vacation day n to activity i. Suppose that{Xn : n = 1, 2, .
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
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. 0 1 0
4. Let {Xn : n = 0, 1, . . .} be a Markov chain with state space {0, 1} and transition probability matrixStarting from 0, find the expected number of transitions until the first visit to 1. P = 2/5
3. Show that the following matrices are the transition probability matrix of the same Markov chain with elements of the state space labeled differently. (2/5 0 0 1/3 1/31 0 3/5 0 1/3 0 1/4 3/4 (2/5
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
2. In Ponza, Italy, a man is stationed at a specific port and can be hired to give sightseeing tours with his boat. If the man is free, it takes an interested tourist a time period, exponentially
1. Consider a parallel system consisting of two components denoted by 1 and 2. Such a system functions if and only if at least one of its components functions. Suppose that each component functions
15. (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,
14. (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
13. (The Yule 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
12. LetN(t) : t ≥ 0be a Poisson process with rate λ. By Example 12.34, the process N(t) : t ≥ 0is a continuous-timeMarkov chain. Hence it satisfies equations (12.12), the Chapman–Kolmogorov
11. 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
10. 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
9. Consider a pure death process with μn = μ, n > 0. For i, j ≥ 0, find pij(t).
8. (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,
7. 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
6. There are m machines in a factory operating independently. The factory has k (k < m)repair persons, and each repair person repairs one machine at a time. Suppose that(i) each machine works for a
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. LetX(t) : t ≥ 0be a continuous-time Markov chain with state space S. Show that for i, j ∈ S and t ≥ 0,In other words, prove Kolmogorov’s backward equations. P(t)=ikPky (t) - Vipij(t). kpi
On a campus building, there are m offices of similar sizes with identical air conditioners. The electrical grid supplies electric energy to the air conditioners whose thermostats turn on and off in
Johnson Medical Associates has two physicians on call, Drs. Dawson and Baick. Dr. Dawson is available to answer patients’ calls for time periods that are exponentially distributed with mean 2
Passengers arrive at a train station according to a Poisson process with rateλ and wait for a train to arrive. Independently, trains arrive at the same station according to a Poisson process with
2. The computers in the Writing Center of a college are inspected at the end of each semester. If a computer needs minor repairs, it will be fixed. If the computer has crashed, it will be replaced
1. On a given day, a retired English professor, Dr. Charles Fish, amuses himself with only one of the following activities: reading (activity 1), gardening (activity 2), or working on his book about
32. For a simple random walk {Xn : n = 0,±1,±2, . . .}, discussed in Examples 12.12 and 12.21, show thatif n + j − i is an even nonnegative integer for whichand it is 0 otherwise. P(X = j | Xoi)
31. In this exercise, we will outline a third technique for solving Example 3.34: We draw cards, one at a time, at random and successively from an ordinary deck of 52 cards with replacement.What is
30. Consider the branching process of Example 12.23. 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
29. Consider the gambler’s ruin problem (Example 3.15) 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
28. 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
27. Let {Xn : n = 0, 1, . . .} be aMarkov 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
26. 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 ∈
25. Show that if P and Q are two transition probability matrices with the same number of rows, and hence columns, then PQis also a transition probability matrix. Note that this implies that if P is a
24. 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
23. For a two-dimensional symmetric random walk, defined in Example 12.21, show that(0, 0) is recurrent and conclude that all states are recurrent.
22. 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 =
21. 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,
20. Let {Xn : n = 0, 1, . . .} be a random walk with state space {0, 1, 2, . . .} and transition probability matrixwhere 0 P = 1 1000 1-p P 0 0 00 1-p 0 P 0 00 1-p 0 P 00 0 1-p 0 p 0 0 0 1-p 1-p0 p
19. Alberto and Angela play backgammon regularly. The probability that Alberto 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
18. 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,
17. In a computerized classroom of a college, let pk > 0 be 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
15. Consider an Ehrenfest chain with 5 balls (see Example 12.14). Find the expected number of balls transferred between two consecutive times that an urn becomes empty.
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
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 theMarkov chain will eventually be
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. (2/5 0
6. Consider an Ehrenfest chain with 5 balls (see Example 12.14). If the probability mass function of X0, the initial number of balls in urn I, is given byfind the probability that, after 6
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, and Xn = 3 if
4. Let {Xn : n = 0, 1, . . .} be a Markov chain with state space {0, 1, 2} and transition probability matrixStarting from 0, what is the probability that the process never enters 1? (1/2 1/4 1/4\
An engineer analyzing a series of digital signals generated by a testing system observes that only 1 out of 15 highly distorted signals follows a highly distorted signal, with no recognizable signal
2. In a Mediterranean restaurant, customers order shish kebab according to a Poisson process with rate λ. If last night, between 7:00 P.M.and 11:00 P.M., eight customers ordered shish kebab dishes,
1. LetN1(t) : t ≥ 0andN2(t) : t ≥ 0be independent Poisson processes with ratesλ and μ, respectively. For n,m ≥ 1, find the probability that the nth event of the process N1(t) : t ≥
6. LetN(t) : t ≥ 0be a Poisson process. For k ≥ 1, let Sk be the time that the kth event occurs. Show that E[Sk | N(t) = n] kt n+1
2. The number of accidents at an intersection is a Poisson processN(t) : t ≥ 0with rate 2.3 per week. Let Xi be the number of injuries in accident i. Suppose that {Xi} is a sequence of independent
1. For a Poisson process with parameter λ, show that, for all ε > 0,as t → ∞. This shows that, for a large t, N(t)/t is a good estimate for λ. P(|N(t) ' 1| ) 0,
Suppose that jobs arrive at a file server at a Poisson rate of 3 per minute. If two jobs arrived within one minute, between 10:00 and 10:01, what is the probability that the first job arrived before
10. A recently built bridge was carefully structured to survive the strengths of the most probable earthquakes. Suppose that (1) the bridge will be maintained properly, and no catastrophe other than
9. Let X be the outcome of a biased die when thrown. Suppose that the probability mass function of X is given byIf the die is thrown 8 times, what is the probability that the sum of the outcomes is
8. For a large positive integer n, let X be a Poisson random variable with parameter n;let Y be a gamma random variable with parameters n and λ, and let W be a binomial random variable with
7. (a) For m ≥ 2, let X1, X2, . . . , Xm be a random sample from a Bernoulli distribution with parameter p. Let ¯X = (X1 + X2 + · · · + Xm)/m. Show that Var( ¯X ) ≤ 1/(4m) ≤ 1/8.(b) For a
6. Let X be a lognormal random variable with parameters μ and σ2. By definition, lnX ∼ N(μ, σ2). In Example 11.6, we showed that for any positive r,Show that even though moments of all orders
5. In a country, which is the target of more natural disasters than most other countries, the time between two consecutive natural disasters is a gamma random variable with parameters r = 3 and λ =
4. For a large n, let X1, X2, . . . , Xn be identically distributed and independent exponential random variables, each with mean 1/5. Find the approximate distribution of the random variable Y = n
3. Let X1, X2, . . . be a sequence of independently selected random numbers from the interval (a, b), a b. For n ≥ 1, let Yn = max(X1,X2, . . . ,Xn). Show that Y converges to the constant random
2. As defined in Exercise 16, Section 7.3, a random variable X is said to be Laplace, if its probability density function is given byLet U and V be independent and identically distributed exponential
1. The probability that next month an ombudsman of a university receives no complaints from a freshman is 0.73. Otherwise, the number of complaints from the freshmen is a normal random variable with
19. 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
18. Show that for a nonnegative random variable X with mean μ, ∀n, nP(X ≥ nμ) ≤ 1.
17. A fair die is rolled 20 times. What is the approximate probability that the sum of the outcomes is between 65 and 75?
16. A randomly selected book from Vernon’s library is X centimeters thick, where X ∼ N(3, 1). Vernon has an empty shelf 87 centimeters long.What is the probability that he can fit 31 randomly
15. In a multiple-choice test with false answers receiving negative scores, the mean of the grades of the students is 0 and its standard deviation is 15. Find an upper bound for the probability that
14. Each time that Ed charges an expense to his credit card, he omits the cents and records only the dollar value. If this month he has charged his credit card 20 times, using Chebyshev’s
13. A psychologistwants 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
12. For a coin, p, the probability of heads is unknown. To estimate p, we flip the coin 5000 times and let bp be the fraction of times it lands heads up. Show that the probability is at least 0.98
11. 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
10. 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 < 1.05).
9. Let X and Y be independent Poisson random variables with parameters λ and μ, respectively.(a) Show that(b) Use part (a) to prove that X + Y is a Poisson random variable with parameter λ + μ. n
8. Suppose that in a community the distributions of heights of men and women (in centimeters)are N(173, 40) and N(160, 20), respectively. Calculate the probability that the average height of 10
7. The moment-generating function of a random variable X is given byFind the moments of X. 1 Mx(t) = t
6. The moment-generating function of X is given byFind P(X > 0). Mx(t) = exp et P(e 21).
5. Let the moment-generating function of a random variable X be given byFind the distribution function of X. Mx(t) = L= (e4/2 (et/2 - e-1/2) 1 ift #0 if t = 0.
4. For a randomvariableX, suppose thatMX(t) = exp(2t2+t). Find E(X) and Var(X).
3. The moment-generating function of a random variable X is given byFind the distribution function of X. Mx(t)= 1 2t be + get. 6 3 1 + 2 IN 3t e
2. The moment-generating function of a random variable X is given byFind Var(X) and P(X ≥ 8). Mx(t)= 23 et) = (1/3 + 1/2 ) 10.
1. Yearly salaries paid to the salespeople employed by a certain company are normally distributed with mean $27,000 and standard deviation $4900. What is the probability that the average wage of a
2. In August of 2017, in the wake of Hurricane Harvey, a group of social workers established an online relief fund, and in only 48 hours, 7000 contributions were made to the fund. Suppose that
1. An actuary wants to determine a new estimate for the average cost, to her insurance company, of the vehicles that are involved in accidents and are considered total losses.Based on the previous
14. 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 k=0 k!
13. Let {X1,X2, . . .} be a sequence of independent standard normal random variables. Let Sn = X2 1 + X2 2 + · · · + X2 n . FindHint: See Example 11.11. lim P(Snn+2n). x+1
12. 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.
11. An investor buys 1000 shares of the XYZ Corporation at $50.00 per share. Subsequently, the stock price varies by $0.125 (1/8) every day, but unfortunately it is just as likely to move down as up.

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