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theory of probability

Questions and Answers of Theory Of Probability

10. Consider two machines. Machine i operates for an exponential time with rate λi and then fails; its repair time is exponential with rate μi , i = 1, 2. The machines act independently of each
9. The birth and death process with parameters λn = 0 and μn = μ, n > 0 is called a pure death process. Find Pij(t).
8. Consider two machines, both of which have an exponential lifetime with mean 1/λ.There is a single repairman that can service machines at an exponential rate μ. Set up the Kolmogorov backward
7. Individuals join a club in accordance with a Poisson process with rate λ. Each new member must pass through k consecutive stages to become a full member of the club.The time it takes to pass
6. Consider a birth and death process with birth rates λi = (i + 1)λ, i 0, and death rates μi = iμ, i 0.(a) Determine the expected time to go from state 0 to state 4.(b) Determine the
5. There are N individuals in a population, some of whom have a certain infection that spreads as follows. Contacts between two members of this population occur in accordance with a Poisson process
4. Potential customers arrive at a single-server station in accordance with a Poisson process with rate λ. However, if the arrival finds n customers already in the station, then he will enter the
3. Consider two machines that are maintained by a single repairman. Machine i functions for an exponential time with rate μi before breaking down, i = 1, 2. The repair times (for either machine) are
2. Suppose that a one-celled organism can be in one of two states—either A or B. An individual in state A will change to state B at an exponential rate α; an individual in state B divides into two
1. A population of organisms consists of both male and female members. In a small colony any particular male is likely to mate with any particular female in any time interval of length h, with
Consider an n-component system where component i, i =1, . . . , n, functions for an exponential time with rate λi and then fails; upon failure, repair begins on component i, with the repair taking
Consider a set of n machines and a single repair facility to service them. Suppose that when machine i, i = 1, . . . , n, goes down it requires an exponentially distributed amount of work with rate
Consider a first come first serve M/M/1 queue, with arrival rateλ and service rate μ, where λ < μ, that is in steady state. Given that customer C spends a total of t time units in the system,
Example 6.6 (A Multiserver Exponential Queueing System) Consider an exponential queueing system in which there are s servers available, each serving at rateμ. An entering customer first waits in
Suppose that customers arrive at a single-server service station in accordance with a Poisson process having rateλ. That is, the times between successive arrivals are independent exponential random
Example 6.4 (A Linear Growth Model with Immigration) A model in whichμn = nμ, n 1λn = nλ + θ, n 0 is called a linear growth process with immigration. Such processes occur naturally in the
Example 6.3 (A Birth Process with Linear Birth Rate) Consider a population whose members can give birth to new members but cannot die. If each member acts independently of the others and takes an
Example 6.2 (The Poisson Process) Consider a birth and death process for whichμn = 0, for all n 0λn = λ, for all n 0 This is a process in which departures never occur, and the time between
97. Consider a conditional Poisson process in which the rate L is, as in Example 5.29, gamma distributed with parameters m and p. Find the conditional density function of L given that N(t) = n.
96. For the conditional Poisson process, let m1 = E[L], m2 = E[L2]. In terms of m1 and m2, find Cov(N(s),N(t)) for s t.
95. Let {N(t), t 0} be a conditional Poisson process with a random rate L.(a) Derive an expression for E[L|N(t) = n].(b) Find, for s > t, E[N(s)|N(t) = n].(c) Find, for s < t, E[N(s)|N(t) = n].
94. A two-dimensional Poisson process is a process of randomly occurring events in the plane such that(i) for any region of area A the number of events in that region has a Poisson distribution with
92. Prove Equation (5.22).
90. In Exercise 89 show that X1 and X2 both have exponential distributions.
89. Some components of a two-component system fail after receiving a shock. Shocks of three types arrive independently and in accordance with Poisson processes.Shocks of the first type arrive at a
88. Customers arrive at the automatic teller machine in accordance with a Poisson process with rate 12 per hour. The amount of money withdrawn on each transaction is a random variable with mean $30
87. Determine Cov[X(t),X(t + s)]when {X(t), t 0} is a compound Poisson process.
86. In good years, storms occur according to a Poisson process with rate 3 per unit time, while in other years they occur according to a Poisson process with rate 5 per unit time. Suppose next year
85. An insurance company pays out claims on its life insurance policies in accordance with a Poisson process having rate λ = 5 per week. If the amount of money paid on each policy is exponentially
84. Let X1,X2, . . . be independent and identically distributed nonnegative continuous random variables having density function f (x).We say that a record occurs at time n if Xn is larger than each
82. Let X1,X2, . . . be independent positive continuous random variables with a common density function f , and suppose this sequence is independent of N, a Poisson random variable with mean λ.
80. Let T1,T2, . . . denote the interarrival times of events of a nonhomogeneous Poisson process having intensity function λ(t).(a) Are the Ti independent?(b) Are the Ti identically distributed?(c)
79. Consider a nonhomogeneous Poisson process whose intensity function λ(t) is bounded and continuous. Show that such a process is equivalent to a process of counted events from a (homogeneous)
78. A store opens at 8 A.M. From 8 until 10 A.M. customers arrive at a Poisson rate of four an hour. Between 10 A.M. and 12 P.M. they arrive at a Poisson rate of eight an hour. From 12 P.M. to 2 P.M.
77. Suppose that customers arrive to a system according to a Poisson process with rate λ. There are an infinite number of servers in this system so a customer begins service upon arrival. The
76. For the model of Example 5.27, find the mean and variance of the number of customers served in a busy period.
75. Suppose that the times between successive arrivals of customers at a single-server station are independent random variables having a common distribution F.Suppose that when a customer arrives, he
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
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
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,
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 S(t) equal the
69. Let {N(t), t 0} be a Poisson process with rate λ. For s < t, find(a) P(N(t) > N(s));(b) P(N(s) = 0,N(t) = 3);(c) E[N(t)|N(s) = 4];(d) E[N(s)|N(t) = 4].
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
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
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, how many lawyers would you
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
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. Let X(t)
61. A system has a random number of flaws that we will suppose is Poisson distributed with meanc. Each of these flaws will, independently, cause the system to fail at a random time having
60. Customers arrive at a bank at a Poisson rate λ. Suppose two customers arrived during the first hour. What is the probability that(a) both arrived during the first 20 minutes?(b) at least one
59. There are two types of claims that are made to an insurance company. Let Ni(t)denote the number of type i claims made by time t, and suppose that {N1(t), t 0}and {N2(t), t 0} are independent
57. Events occur according to a Poisson process with rate λ = 2 per hour.(a) What is the probability that no event occurs between 8 P.M. and 9 P.M.?(b) Starting at noon, what is the expected time at
56. An event independently occurs on each day with probability p. Let N(n) denote the total number of events that occur on the first n days, and let Tr denote the day on which the rth event
55. Consider a single server queuing system where customers arrive according to a Poisson process with rate λ, service times are exponential with rate μ, and customers are served in the order of
54. A viral linear DNA molecule of length, say, 1 is often known to contain a certain“marked position,” with the exact location of this mark being unknown. One approach to locating the marked
53. The water level of a certain reservoir is depleted at a constant rate of 1000 units daily. The reservoir is refilled by randomly occurring rainfalls. Rainfalls occur according to a Poisson
52. Teams 1 and 2 are playing a match. The teams score points according to independent Poisson processes with respective rates λ1 and λ2. If the match ends when one of the teams has scored k more
51. If an individual has never had a previous automobile accident, then the probability he or she has an accident in the next h time units is βh + o(h); on the other hand, if he or she has ever had
50. The number of hours between successive train arrivals at the station is uniformly distributed on (0, 1). Passengers arrive according to a Poisson process with rate 7 per hour. Suppose a train has
49. Events occur according to a Poisson process with rate λ. Each time an event occurs, we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to
48. Consider an n-server parallel queuing system where customers arrive according to a Poisson process with rate λ, where the service times are exponential random variables with rate μ, and where
47. Consider a two-server parallel queuing system where customers arrive according to a Poisson process with rate λ, and where the service times are exponential with rateμ. Moreover, suppose that
45. Let {N(t), t 0} be a Poisson process with rate λ that is independent of the nonnegative random variable T with mean μ and variance σ2. Find(a) Cov(T, N(T)),(b) Var(N(T)).
44. Cars pass a certain street location according to a Poisson process with rate λ.A woman who wants to cross the street at that location waits until she can see that no cars will come by in the
43. Customers arrive at a two-server service station according to a Poisson process with rate λ. Whenever a new customer arrives, any customer that is in the system immediately departs. A new
42. Let {N(t), t 0} be a Poisson process with rate λ. Let Sn denote the time of the nth event. Find(a) E[S4],(b) E[S4|N(1) = 2],(c) E[N(4) − N(2)|N(1) = 3].
41. In Exercise 40 what is the probability that the first event of the combined process is from the N1 process?
40. Show that if {Ni(t), t 0} are independent Poisson processes with rate λi , i = 1, 2, then {N(t), t 0} is a Poisson process with rate λ1 + λ2 where N(t) = N1(t) +N2(t).
39. A certain scientific theory supposes that mistakes in cell division occur according to a Poisson process with rate 2.5 per year, and that an individual dies when 196 such mistakes have occurred.
38. Let {Mi(t), t 0}, i = 1, 2, 3 be independent Poisson processes with respective ratesλi , i = 1, 2, and set N1(t) = M1(t) + M2(t), N2(t) = M2(t) + M3(t)The stochastic process {(N1(t),N2(t)), t
37. A machine works for an exponentially distributed time with rate μ and then fails. A repair crew checks the machine at times distributed according to a Poisson process with rate λ; if the
35. Show that Definition 5.1 of a Poisson process implies Definition 5.3.
34. Two individuals, A and B, both require kidney transplants. If she does not receive a new kidney, then A will die after an exponential time with rate μA, and B after an exponential time with rate
32. Let X be a uniform random variable on (0, 1), and consider a counting process where events occur at times X + i, for i = 0, 1, 2, . . ..(a) Does this counting process have independent
31. A doctor has scheduled two appointments, one at 1 P.M. and the other at 1:30 P.M.The amounts of time that appointments last are independent exponential random variables with mean 30 minutes.
30. The lifetimes of A’s dog and cat are independent exponential random variables with respective rates λd and λc. One of them has just died. Find the expected additional lifetime of the other
28. Consider n components with independent lifetimes, which are such that component i functions for an exponential time with rate λi . Suppose that all components are initially in use and remain so
27. Show, in Example 5.7, that the distributions of the total cost are the same for the two algorithms.
26. Each entering customer must be served first by server 1, then by server 2, and finally by server 3. The amount of time it takes to be served by server i is an exponential random variable with
25. Customers can be served by any of three servers, where the service times of server i are exponentially distributed with rate μi , i = 1, 2, 3. Whenever a server becomes free, the customer who
24. There are two servers available to process n jobs. Initially, each server begins work on a job. Whenever a server completes work on a job, that job leaves the system and the server begins
23. A flashlight needs two batteries to be operational. Consider such a flashlight along with a set of n functional batteries—battery 1, battery 2, . . . , battery n. Initially, battery 1 and 2 are
22. Suppose in Exercise 21 you arrive to find two others in the system, one being served by server 1 and one by server 2. What is the expected time you spend in the system?Recall that if server 1
21. In a certain system, a customer must first be served by server 1 and then by server 2.The service times at server i are exponential with rate μi , i = 1, 2. An arrival finding server 1 busy
20. Consider a two-server system in which a customer is served first by server 1, then by server 2, and then departs. The service times at server i are exponential random variables with rates μi , i
19. Repeat Exercise 18, but this time suppose that the Xi are independent exponentials with respective rates μi , i = 1, 2.
18. Let X1 and X2 be independent exponential random variables, each having rate μ. Let X(1) = minimum(X1,X2) and X(2) = maximum(X1,X2)Find (a) E[X(1)], (b) Var[X(1)], (c) E[X(2)], (d) Var[X(2)].
17. A set of n cities is to be connected via communication links. The cost to construct a link between cities i and j is Cij , i = j. Enough links should be constructed so that for each pair of
15. One hundred items are simultaneously put on a life test. Suppose the lifetimes of the individual items are independent exponential random variables with mean 200 hours. The test will end when
13. Find, in Example 5.10, the expected time until the nth person on line leaves the line(either by entering service or departing without service).
12. If Xi , i = 1, 2, 3, are independent exponential random variables with rates λi , i = 1, 2, 3, find(a) P{X1 < X2 < X3},(b) P{X1 < X2| max(X1, X2, X3) = X3},(c) E[maxXi |X1 < X2 < X3],(d)
11. Let X, Y1, . . . ,Yn be independent exponential random variables; X having rate λ, and Yi having rate μ. Let Aj be the event that the jth smallest of these n + 1 random variables is one of the
10. Let X and Y be independent exponential random variables with respective rates λand μ. Let M = min(X,Y). Find(a) E[MX|M = X],(b) E[MX|M = Y],(c) Cov(X,M).
9. Machine 1 is currently working. Machine 2 will be put in use at a time t from now. If the lifetime of machine i is exponential with rate λi , i = 1, 2, what is the probability that machine 1 is
5. The lifetime of a radio is exponentially distributed with a mean of ten years. If Jones buys a ten-year-old radio, what is the probability that it will be working after an additional ten years?
4. Consider a post office with two clerks. Three people, A, B, and C, enter simultaneously.A and B go directly to the clerks, and C waits until either A or B leaves before he begins service. What is
3. Let X be an exponential random variable. Without any computations, tell which one of the following is correct. Explain your answer.(a) E[X2|X > 1] = E[(X + 1)2](b) E[X2|X > 1] = E[X2] + 1(c)
2. Suppose that you arrive at a single-teller bank to find five other customers in the bank, one being served and the other four waiting in line. You join the end of the line. If the service times
1. The time T required to repair a machine is an exponentially distributed random variable with mean 12(hours).(a) What is the probability that a repair time exceeds 12 hour?(b) What is the
An insurance company feels that each of its policyholders has a rating value and that a policyholder having rating value λ will make claims at times distributed according to a Poisson process with
In Example 5.26, find the approximate probability that at least 240 people migrate to the area within the next 50 weeks

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