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introduction to probability statistics

Introduction To Probability Models 12th Edition Sheldon M Ross - Solutions

16. If is a Martingale, show that
15. The current price of a stock is 100. Suppose that the logarithm of the price of the stock changes according to a Brownian motion process with drift coefficient and variance parameter Give the Black–Scholes cost of an option to buy the stock at time 10 for a cost of (a) 100 per unit.(b) 120
14. The present price of a stock is 100. The price at time 1 will be either 50, 100, or 200. An option to purchase y shares of the stock at time 1 for the (present value) price ky costs cy.(a) If , show that an arbitrage opportunity occurs if and only if .(b) If , show that there is not an
13. Verify the statement made in the remark following Example 10.2.
12. A stock is presently selling at a price of $50 per share. After one time period, its selling price will (in present value dollars) be either$150 or $25. An option to purchase y units of the stock at time 1 can be purchased at cost cy.(a) What should c be in order for there to be no sure win?(b)
11. Consider a process whose value changes every h time units; its new value being its old value multiplied either by the factor with probability , or by the factor with probability . As h goes to zero, show that this process converges to geometric Brownian motion with drift coefficient μand
*10. Let be a Brownian motion process with drift coefficient μ and variance parameter . What is the conditional distribution of given that when(a) ?(b) ?
9. Let be a Brownian motion process with drift coefficientμ and variance parameter . What is the joint density function of and ?
8. Consider the random walk that in each Δt time unit either goes up or down the amount with respective probabilities p and , where .(a) Argue that as the resulting limiting process is a Brownian motion process with drift rate μ.(b) Using part (a) and the results of the gambler's ruin problem
7. Compute an expression for
6. Suppose you own one share of a stock whose price changes according to a standard Brownian motion process. Suppose that you purchased the stock at a price , and the present price is b.You have decided to sell the stock either when it reaches the price or when an additional time t goes by
*5. What is ?
4. Show that
*3. Compute for .
2. Compute the conditional distribution of given that and, where .
*1. What is the distribution of , ?
15. A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers,(a) what fraction of potential customers enter the system?(b) what would
*33. Consider two queues with respective parameters. Suppose they share a common waiting room that can hold at most three customers. That is, whenever an arrival finds her server busy and three customers in the waiting room, she goes away. Find the limiting probability that there will be n queue 1
34. Four workers share an office that contains four telephones. At any time, each worker is either “working” or “on the phone.” Each“working” period of worker i lasts for an exponentially distributed time with rate , and each “on the phone” period lasts for an exponentially
35. Consider a time reversible continuous-time Markov chain having infinitesimal transition rates and limiting probabilities . Let A denote a set of states for this chain, and consider a new continuoustime Markov chain with transition rates given by where c is an arbitrary positive number. Show
36. Consider a system of n components such that the working times of component , are exponentially distributed with rate .When a component fails, however, the repair rate of component i depends on how many other components are down. Specifically, suppose that the instantaneous repair rate of
37. A hospital accepts k different types of patients, where type i patients arrive according to a Poisson proccess with rate , with these k Poisson processes being independent. Type i patients spend an exponentially distributed length of time with rate in the hospital, . Suppose that each type i
38. Consider an n server system where the service times of server i are exponentially distributed with rate . Suppose customers arrive in accordance with a Poisson process with rate λ, and that an arrival who finds all servers busy does not enter but goes elsewhere. Suppose that an arriving
39. Suppose in Exercise 38 that an entering customer is served by the server who has been idle the shortest amount of time.(a) Define states so as to analyze this model as a continuous-time Markov chain.(b) Show that this chain is time reversible.(c) Find the limiting probabilities.
*40. Consider a continuous-time Markov chain with statesI mage,which spends an exponential time with rate in state i during each visit to that state and is then equally likely to go to any of the other states.(a) Is this chain time reversible?(b) Find the long-run proportions of time it spends in
41. Show in Example 6.22 that the limiting probabilities satisfy Eqs.(6.33), (6.34), and (6.35).
42. In Example 6.22 explain why we would have known before analyzing Example 6.22 that the limiting probability there are j customers with server i is . (What we would not have known was that the number of customers at the two servers would, in steady state, be independent.)
43. Consider a sequential queueing model with three servers, where customers arrive at server 1 in accordance with a Poisson process with rate λ. After completion at server 1 the customer then moves to server 2; after a service completion at server 2 the customer moves to server 3; after a service
44. A system of N machines operates as follows. Each machine works for an exponentially distributed time with rate λ before failing.Upon failure, a machine must go through two phases of service.Phase 1 service lasts for an exponential time with rate μ, and there are always servers available for
45. For the continuous-time Markov chain of Exercise 3 present a uniformized version.
46. In Example 6.24, we computed , the expected occupation time in state 0 by time t for the two-state continuoustime Markov chain starting in state 0. Another way of obtaining this quantity is by deriving a differential equation for it.(a) Show that(b) Show that(c) Solve for .
47. Let be the occupation time for state 0 in the two-state continuous-time Markov chain. Find .
32. Customers arrive at a two-server station in accordance with a Poisson process having rate λ. Upon arriving, they join a single queue. Whenever a server completes a service, the person first in line enters service. The service times of server i are exponential with rate , whereI mage. An
31. A total of N customers move about among r servers in the following manner. When a customer is served by server i, he then goes over to server j, , with probability . If the server he goes to is free, then the customer enters service; otherwise he joins the queue. The service times are all
*16. The following problem arises in molecular biology. The surface of a bacterium consists of several sites at which foreign molecules—some acceptable and some not—become attached. We consider a particular site and assume that molecules arrive at the site according to a Poisson process with
17. Each time a machine is repaired it remains up for an exponentially distributed time with rate λ. It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to repair the machine is exponential with rate ; if it is a type 2 failure, then the repair time is
18. After being repaired, a machine functions for an exponential time with rate λ and then fails. Upon failure, a repair process begins. The repair process proceeds sequentially through k distinct phases. First a phase 1 repair must be performed, then a phase 2, and so on. The times to complete
*19. A single repairperson looks after both machines 1 and 2. Each time it is repaired, machine i stays up for an exponential time with rate , 2. When machine i fails, it requires an exponentially distributed amount of work with rate to complete its repair. The repairperson will always service
20. There are two machines, one of which is used as a spare. A working machine will function for an exponential time with rate λand will then fail. Upon failure, it is immediately replaced by the other machine if that one is in working order, and it goes to the repair facility. The repair facility
21. Suppose that when both machines are down in Exercise 20 a second repairperson is called in to work on the newly failed one.Suppose all repair times remain exponential with rate μ. Now find the proportion of time at least one machine is working, and compare your answer with the one obtained in
22. Customers arrive at a single-server queue in accordance with a Poisson process having rate λ. However, an arrival that finds n customers already in the system will only join the system with probabilityI mage. That is, with probabilityI mage such an arrival will not join the system. Show that
23. A job shop consists of three machines and two repairmen. The amount of time a machine works before breaking down is exponentially distributed with mean 10. If the amount of time it takes a single repairman to fix a machine is exponentially distributed with mean 8, then(a) what is the average
*24. Consider a taxi station where taxis and customers arrive in accordance with Poisson processes with respective rates of one and two per minute. A taxi will wait no matter how many other taxis are present. However, an arriving customer that does not find a taxi waiting leaves. Find(a) the
25. Customers arrive at a service station, manned by a single server who serves at an exponential rate , at a Poisson rate λ. After completion of service the customer then joins a second system where the server serves at an exponential rate . Such a system is called a tandem or sequential queueing
26. Consider an ergodic queue in steady state (that is, after a long time) and argue that the number presently in the system is independent of the sequence of past departure times. That is, for instance, knowing that there have been departures 2, 3, 5, and 10 time units ago does not affect the
27. In the queue if you allow the service rate to depend on the number in the system (but in such a way so that it is ergodic), what can you say about the output process? What can you say when the service rate μ remains unchanged butI mage?
*28. If andI mage are independent continuous-time Markov chains, both of which are time reversible, show that the process
29. Consider a set of n machines and a single repair facility to service these machines. Suppose that when machine , fails it requires an exponentially distributed amount of work with rate to repair it. The repair facility divides its efforts equally among all failed machines in the sense that
30. Consider a graph with nodesI mage and theI mage arcsI mage. (See Section 3.6.2 for appropriate definitions.) Suppose that a particle moves along this graph as follows: Events occur along the arcs Image according to independent Poisson processes with ratesI mage.An event along arcI mage causes
48. Consider the two-state continuous-time Markov chain. Starting in state 0, find .
49. Let Y denote an exponential random variable with rate λ that is independent of the continuous-time Markov chain and let(a) Show that where is 1 when and 0 when .(b) Show that the solution of the preceding set of equations is given by where is the matrix of elements , I is the identity matrix,
*50.(a) Show that Approximation 1 of Section 6.9 is equivalent to uniformizing the continuous-time Markov chain with a value v such that and then approximating by .(b) Explain why the preceding should make a good approximation.Hint: What is the standard deviation of a Poisson random variable with
*18. Compute the renewal function when the interarrival distribution F is such that
19. For the renewal process whose interarrival times are uniformly distributed over (0, 1), determine the expected time fromI mage until the next renewal.
20. For a renewal reward process consider whereI mage represents the average reward earned during the first n cycles. Show thatI mage asI mage.
21. Consider a single-server bank for which customers arrive in accordance with a Poisson process with rate λ. If a customer will enter the bank only if the server is free when he arrives, and if the service time of a customer has the distribution G, then what proportion of time is the server busy?
*22. J's car buying policy is to always buy a new car, repair all breakdowns that occur during the first T time units of ownership, and then junk the car and buy a new one at the first breakdown that occurs after the car has reached age T. Suppose that the time until the first breakdown of a new
23. In a serve and rally competition involving players A and B, each rally that begins with a serve by player A is won by player A with probabilityI mage and is won by player B with probabilityI mage, whereas each rally that begins with a serve by player B is won by player A with probabilityI mage
24. Wald's equation can also be proved by using renewal reward processes. Let N be a stopping time for the sequence of independent and identically distributed random variables .(a) Let Image. Argue that the sequence of random variables Image is independent of Image and has the same distribution as
25. Suppose in Example 7.15 that the arrival process is a Poisson process and suppose that the policy employed is to dispatch the train every t time units.(a) Determine the average cost per unit time.(b) Show that the minimal average cost per unit time for such a policy is approximately Image plus
26. Consider a train station to which customers arrive in accordance with a Poisson process having rate λ. A train is summoned whenever there are N customers waiting in the station, but it takes K units of time for the train to arrive at the station. When it arrives, it picks up all waiting
27. A machine consists of two independent components, the ith of which functions for an exponential time with rateI mage. The machine functions as long as at least one of these components function. (That is, it fails when both components have failed.)When a machine fails, a new machine having both
28. In Example 7.17, what proportion of the defective items produced is discovered?
29. Consider a single-server queueing system in which customers arrive in accordance with a renewal process. Each customer brings in a random amount of work, chosen independently according to the distribution G. The server serves one customer at a time.However, the server processes work at rate i
*30. For a renewal process, let be the age at time t. Prove that if Image, then with probability 1
31. If and are, respectively, the age and the excess at time t of a renewal process having an interarrival distribution F, calculate
32. Determine the long-run proportion of time thatI mage.
17. In Example 7.6, suppose that potential customers arrive in accordance with a renewal process having interarrival distribution F. Would the number of events by time t constitute a (possibly delayed) renewal process if an event corresponds to a customer(a) entering the bank?(b) leaving the
16. A deck of 52 playing cards is shuffled and the cards are then turned face up one at a time. Let equal 1 if the ith card turned over is an ace, and let it be 0 otherwise,I mage. Also, let N denote the number of cards that need be turned over until all four aces appear. That is, the final ace
1. Is it true that(a) Image if and only if Image?(b) Image if and only if Image?(c) Image if and only if Image?
2. Suppose that the interarrival distribution for a renewal process is Poisson distributed with mean μ. That is, suppose(a) Find the distribution of .(b) Calculate Image.
3. Let andI mage be independent renewal processes. Let Image.(a) Are the interarrival times of independent?(b) Are they identically distributed?(c) Is a renewal process?
4. LetI mage be independent uniform (0, 1) random variables, and define N by Image What is ?
*5. Consider a renewal process having a gammaI mage interarrival distribution. That is, the interarrival density is Image(a) Show that Image (b) Show that Image whereI mage is the largest integer less than or equal toI mage.Hint: Use the relationship between the gammaI mage distribution and the sum
6. Two players are playing a sequence of games, which begin when one of the players serves. Suppose that player 1 wins each game she serves with probabilityI mage and wins each game her opponent serves with probabilityI mage. Further, suppose that the winner of a game becomes the server of the next
7. Mr. Smith works on a temporary basis. The mean length of each job he gets is three months. If the amount of time he spends between jobs is exponentially distributed with mean 2, then at what rate does Mr. Smith get new jobs?
*8. A machine in use is replaced by a new machine either when it fails or when it reaches the age of T years. If the lifetimes of successive machines are independent with a common distribution F having densityf, show that(a) the long-run rate at which machines are replaced equals(b) the long-run
9. A worker sequentially works on jobs. Each time a job is completed,a new one is begun. Each job, independently, takes a random amount of time having distribution F to complete. However, independently of this, shocks occur according to a Poisson process with rate λ. Whenever a shock occurs, the
10. Consider a renewal process with mean interarrival time μ.Suppose that each event of this process is independently “counted”with probability p. LetI mage denote the number of counted events by timeI mage.(a) Is Image a renewal process?(b) What is Image?
11. Events occur according to a Poisson process with rate λ. Any event that occurs within a time d of the event that immediately preceded it is called a d-event. For instance, ifI mage and events occur at timesI mage then the events at times 2.8 and 6.6 would be d-events.(a) At what rate do
12. LetI mage be independent uniform random variables. Let and letI mage.(a) Find by conditioning on the value of .(b) Find by conditioning on N.(c) Find by using Wald's equation.
13. In each game played one is equally likely to either win or lose 1.Let X be your cumulative winnings if you use the strategy that quits playing if you win the first game, and plays two more games and then quits if you lose the first game.(a) Use Wald's equation to determine .(b) Compute the
14. Consider the gambler's ruin problem where on each bet the gambler either wins 1 with probability p or loses 1 with probability Image. The gambler will continue to play until his winnings are eitherI mage or −i. (That is, starting with i the gambler will quit when his fortune reaches either N
15. Consider a miner trapped in a room that contains three doors.Door 1 leads him to freedom after two days of travel; door 2 returns him to his room after a four-day journey; and door 3 returns him to his room after a six-day journey. Suppose at all times he is equally likely to choose any of the
33. In Example 7.16, find the long-run proportion of time that the server is busy.
45. LetI mage be a Poisson process with rate λ that is independent of the nonnegative random variable T with mean μ and variance .Find(a) ,(b) .
63. Consider an infinite server queuing system in which customers arrive in accordance with a Poisson process with rate λ, and where the service distribution is exponential with rate μ. LetI mage denote the number of customers in the system at time t. Find(a) ;(b) .Hint: Divide the customers in
*64. Suppose that people arrive at a bus stop in accordance with a Poisson process with rate λ. The bus departs at time t. Let X denote the total amount of waiting time of all those who get on the bus at time t. We want to determine . LetI mage denote the number of arrivals by time t.(a) What is
65. An average of 500 people pass the California bar exam each year.A California lawyer practices law, on average, for 30 years.Assuming these numbers remain steady, roughly how many lawyers would you expect California to have in 2050?
66. Policyholders of a certain insurance company have accidents at times distributed according to a Poisson process with rate λ. The amount of time from when the accident occurs until a claim is made has distribution G.(a) Find the probability there are exactly n incurred but as yet unreported
67. Satellites are launched into space at times distributed according to a Poisson process with rate λ. Each satellite independently spends a random time (having distribution G) in space before falling to the ground. Find the probability that none of the satellites in the air at time t was
68. Suppose that electrical shocks having random amplitudes occur at times distributed according to a Poisson processI mage with rate λ.Suppose that the amplitudes of the successive shocks are independent both of other amplitudes and of the arrival times of shocks, and also that the amplitudes
69. Suppose in Example 5.19 that a car can overtake a slower moving car without any loss of speed. Suppose a car that enters the road at time s has a free travel time equal to . Find the distribution of the total number of other cars that it encounters on the road (either by passing or by being
70. For the infinite server queue with Poisson arrivals and general service distribution G, find the probability that(a) the first customer to arrive is also the first to depart.Let equal the sum of the remaining service times of all customers in the system at time t.(b) Argue that is a compound
71. LetI mage be a Poisson process with rateI mage.(a) Find .(b) Find .(c) Find .
72. A cable car starts off with n riders. The times between successive stops of the car are independent exponential random variables with rate λ. At each stop one rider gets off. This takes no time, and no additional riders get on. After a rider gets off the car, he or she walks home.
73. Shocks occur according to a Poisson process with rate λ, and each shock independently causes a certain system to fail with probability p. Let T denote the time at which the system fails and let N denote the number of shocks that it takes.(a) Find the conditional distribution of T given that
74. The number of missing items in a certain location, call it X, is a Poisson random variable with mean λ. When searching the location, each item will independently be found after an exponentially distributed time with rate μ. A reward of R is received for each item found, and a searching cost
75. If are independent exponential random variables with rate λ, find(a) ;(b) , .Hint: Interpret as the interarrival times of a Poisson process.
76. For the model of Example 5.27, find the mean and variance of the number of customers served in a busy period.

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