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theory of probability

Questions and Answers of Theory Of Probability

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
58. Consider the coupon collecting problem where there are m distinct types of coupons, and each new coupon collected is type j with probability pj , m j=1 pj = 1. Suppose you stop collecting when
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
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
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
62. Suppose that the number of typographical errors in a new text is Poisson distributed with mean λ. Two proofreaders independently read the text. Suppose that each error is independently found by
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
75. Suppose that the times between successive arrivals of customers at a singleserver station are independent random variables having a common distribution F.Suppose that when a customer arrives, he
87. Determine Cov[X(t),X(t + s)]when {X(t),t 0} is a compound Poisson process.
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 The amount of money withdrawn on each transaction is a random variable with mean $30
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
90. In Exercise 89 show that X1 and X2 both have exponential distributions.
91. Let X1,X2,...,Xn be independent and identically distributed exponential random variables. Show that the probability that the largest of them is greater than the sum of the others is n/2n−1.
92. Prove Equation (5.22).
93. Prove that(a) max(X1,X2) = X1 + X2 − min(X1,X2) and, in general,(b) max(X1,...,Xn) =n 1Xi −i
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
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
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
76. For the model of Example 5.27, find the mean and variance of the number of customers served in a busy period.
77. Events occur according to a nonhomogeneous Poisson process whose mean value function is given by m(t) = t 2 + 2t, t 0 What is the probability that n events occur between times t = 4 and t = 5?
78. A store opens at 8 A.M. From 8 until 10 customers arrive at a Poisson rate of four an hour. Between 10 and 12 they arrive at a Poisson rate of eight an hour.From 12 to 2 the arrival rate
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 ?
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)
81. (a) Let {N(t),t 0} be a nonhomogeneous Poisson process with mean value function m(t). Given N(t) = n, show that the unordered set of arrival times has the same distribution as n independent and
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. Define
83. Suppose that {N0(t), t 0} is a Poisson process with rate λ = 1. Let λ(t)denote a nonnegative function of t, and let m(t) = t 0λ(s)ds Define N(t) by N(t) = N0(m(t))Argue that {N(t),t 0} is a
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 of
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.
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
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 waits
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
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
24. There are 2 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
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 has
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
27. Show, in Example 5.7, that the distributions of the total cost are the same for the two algorithms.
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
29. Let X and Y be independent exponential random variables with respective rates λ and μ, where λ>μ. Let c > 0.(a) Show that the conditional density function of X, given that X + Y =c, is fX|X+Y
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.
10. Let X and Y be independent exponential random variables with respective rates λ and μ. Let M = min(X,Y). Find(a) E[MX|M = X],(b) E[MX|M = Y],(c) Cov(X,M).
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 j th smallest of these n + 1 random variables is one of the Yi.
12. If Xi, i = 1, 2, 3, are independent exponential random variables with ratesλi, 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) E[maxXi].
13. Find, in Example 5.8, the expected time until the nth person on line leaves the line (either by entering service or departing without service).
14. Let X be an exponential random variable with rate λ.(a) Use the definition of conditional expectation to determine E[X|X
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
16. Suppose in Example 5.3 that the time it takes server i to serve customers is exponentially distributed with mean 1/λi, i = 1, 2. What is the expected time until all three customers have left the
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
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)].
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
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.
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
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
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 )).
46. Let {N(t), t 0} be a Poisson process with rate λ, that is independent of the sequence X1, X2,... of independent and identically distributed random variables with mean μ and variance σ2. Find
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 rate μ. Moreover, suppose that
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
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
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
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
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?
32. There are three jobs and a single worker who works first on job 1, then on job 2, and finally on job 3. The amounts of time that he spends on each job are independent exponential random variables
33. Let X and Y be independent exponential random variables with respective rates λ and μ.(a) Argue that, conditional on X>Y , the random variables min(X,Y) and X − Y are independent.(b) Use part
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
35. Show that Definition 5.1 of a Poisson process implies Definition 5.3.
36. Let S(t) denote the price of a security at time t. A popular model for the process {S(t),t 0} supposes that the price remains unchanged until a “shock”occurs, at which time the price is
37. Cars cross a certain point in the highway in accordance with a Poisson process with rate λ = 3 per minute. If Reb blindly runs across the highway, then what is the probability that she will be
38. Let {Mi(t),t 0}, i = 1, 2 be independent Poisson processes with respective rates λi, 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 0} is
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.
40. Show that if {Ni(t),t 0} are independent Poisson processes with rateλi, i = 1, 2, then {N(t),t 0} is a Poisson process with rate λ1 + λ2 where N(t) = N1(t) + N2(t).
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
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
48. Sixty percent of the families in a certain community own their own car, thirty percent own their own home, and twenty percent own both their own car and their own home. If a family is randomly
47. For a fixed event B, show that the collection P(A|B), defined for all events A, satisfies the three conditions for a probability. Conclude from this that P(A|B) = P(A|BC)P(C|B) + P(A|BCc)P(Cc|B)
46. Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which
45. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn c additional balls of the same color are put in with it. Now suppose that we
44. Urn 1 has five white and seven black balls. Urn 2 has three white and twelve black balls. We flip a fair coin. If the outcome is heads, then a ball from urn 1 is selected, while if the outcome is
43. Suppose we have ten coins which are such that if the ith one is flipped then heads will appear with probability i/10, i = 1, 2,..., 10. When one of the coins is randomly selected and flipped, it
42. There are three coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time.When one of the three coins is selected
41. In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling.(a) What is the probability that this rat is a pure black rat (as
40. (a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is the fair coin?
39. Stores A, B, and C have 50, 75, and 100 employees, and, respectively, 50, 60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One
38. Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2. A ball is then drawn from
37. In Exercise 36, what is the probability that the first box was the one selected given that the marble is white?
36. Consider two boxes, one containing one black and one white marble, the other, two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box.
35. A fair coin is continually flipped. What is the probability that the first four flips are(a) H, H, H, H?(b) T , H, H, H?(c) What is the probability that the pattern T , H, H, H occurs before the
34. Mr. Jones has devised a gambling system for winning at roulette. When he bets, he bets on red, and places a bet only when the ten previous spins of the roulette have landed on a black number. He
33. In a class there are four freshman boys, six freshman girls, and six sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at
32. Suppose all n men at a party throw their hats in the center of the room.Each man then randomly selects a hat. Show that the probability that none of the n men selects his own hat is 12!− 1 3!+1
31. What is the conditional probability that the first die is six given that the sum of the dice is seven?
30. Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability 0.7, whereas George, independently, hits the target with
29. Suppose that P(E) = 0.6. What can you say about P(E|F) when(a) E and F are mutually exclusive?(b) E ⊂ F?(c) F ⊂ E?
28. If the occurrence of B makes A more likely, does the occurrence of A make B more likely?
27. Suppose in Exercise 26 we had defined the events Ei, i = 1, 2, 3, 4, by E1 = {one of the piles contains the ace of spades}, E2 = {the ace of spades and the ace of hearts are in different piles},
26. A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events E1,E2,E3, and E4 as follows:E1 = {the first pile has exactly 1 ace}, E2 = {the
25. Two cards are randomly selected from a deck of 52 playing cards.(a) What is the probability they constitute a pair (that is, that they are of the same denomination)?(b) What is the conditional
24. In an election, candidate A receives n votes and candidate B receives m votes, where n>m. Assume that in the count of the votes all possible orderings of the n + m votes are equally likely. Let
23. For events E1,E2,...,En show that P(E1E2 ···En) = P(E1)P(E2|E1)P(E3|E1E2)···P(En|E1 ···En−1)

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