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theory of probability

Questions and Answers of Theory Of Probability

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
Consider a continuous-time Markov chain with states 1,..., n, which spends an exponential time with rate vi in state i during each visit to that state and is then equally likely to go to any of the
Show in Example 6.22 that the limiting probabilities satisfy Equations (6.33),(6.34), and (6.35).
In Example 6.22 explain why we would have known before analyzing Example 6.22 that the limiting probability there are j customers with serveri is(λ/μi)j(1−λ/μi), i = 1, 2, j 0. (What we would
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
For the continuous-time Markov chain of Exercise 3 present a uniformized version.
In Example 6.20, we computed m(t) = E[O(t)], the expected occupation time in state 0 by time t for the two-state continuous-time Markov chain starting in state 0. Another way of obtaining this
Let O(t) be the occupation time for state 0 in the two-state continuous-time Markov chain. Find E[O(t)|X(0) = 1].
Consider the two-state continuous-time Markov chain. Starting in state 0, find Cov[X(s), X(t)].
Let Y denote an exponential random variable with rate λ that is independent of the continuous-time Markov chain {X(t)} and let P¯i j = P{X(Y ) = j|X(0) = i}(a) Show that P¯i j = 1 vi + λk qik
(a) Show that Approximation 1 of Section 6.9 is equivalent to uniformizing the continuous-time Markov chain with a value v such that vt = n and then approximating Pi j(t) by P∗n i j .(b) Explain
Let X be the time between the first and the second event of a Hawkes process with mark distribution F. Find P(X > t).
Let M(t) = E[D(t)] in Example 5.21.(a) Show that M(t + h) = M(t) + e−αtλhμ + o(h)(b) Use (a) to show that M(t) = λμe−αt(c) Show that M(t) = λμα(1 − e−αt)
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.
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.
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].
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 mean
Prove that(a) max(X1, X2) = X1 + X2 − min(X1, X2) and, in general,(b) max(X1,..., Xn) = n 1Xi −i< j min(Xi, X j)
Prove Equation (5.22).
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. That
In Exercise 89 show that X1 and X2 both have exponential distributions.
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
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 and
Determine Cov[X(t), X(t + s)]when {X(t), t 0} is a compound Poisson process?
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 will
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
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 the
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
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
(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
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) Find
Suppose that events occur according to a nonhomogeneous Poisson process with intensity function λ(t), t > 0. Further, suppose that an event that occurs at time s is a type 1 event with probability
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. the
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?
For the model of Example 5.27, find the mean and variance of the number of customers served in a busy period.
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 or
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
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
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
Let Sn denote the time of the nth event of the Poisson process {N(t), t 0} having rate λ. Show, for an arbitrary function g, that the random variable N(t)i=1 g(Si)has the same distribution as the
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 sum of
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 t0. Find the distribution
Suppose that electrical shocks having random amplitudes occur at times distributed according to a Poisson process {N(t), t 0} with rate λ. Suppose that the amplitudes of the successive shocks are
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
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
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 expect
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
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) denote
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
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 distribution G.
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 arrived
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 Poisson
Each round played by a contestant is either a success with probability p or a failure with probability 1− p. If the round is a success, then a random amount of money having an exponential
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
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 occurs.(a)
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 their
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
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 process
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
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 a
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
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 some
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 any
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
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
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 )).
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 next T
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 arrival
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],
In Exercise 40 what is the probability that the first event of the combined process is from the N1 process?
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).
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.
Let {Mi(t), t 0},i = 1, 2, 3 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
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 machine is
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
If {N(t), t 0} is a Poisson process with rate λ, verify that {Ns(t), t 0}satisfies the axioms for being a Poisson process with rate λ, where Ns(t) =N(s + t) − N(s).
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 rateμA, and B after an exponential time with rate μB.
Let X and Y be independent exponential random variables with respective ratesλ and μ.(a) Argue that, conditional on X > Y , the random variables min(X, Y ) and X −Y are independent.(b) Use part
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 increments?(b)
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. Assuming
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 pet.
Let X and Y be independent exponential random variables with respective ratesλ 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
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
Show, in Example 5.7, that the distributions of the total cost are the same for the two algorithms.
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 rate
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
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 processing
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
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
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 in
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 = 1,
In a mile race between A and B, the time it takes A to complete the mile is an exponential random variable with rate λa and is independent of the time it takes B to complete the mile, which is an
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)].
A set of n cities is to be connected via communication links. The cost to construct a link between cities i and j is Ci j,i = j. Enough links should be constructed so that for each pair of cities
There are three jobs that need to be processed, with the processing time of job i being exponential with rate μi . There are two processors available, so processing on two of the jobs can
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 there
I am waiting for two friends to arrive at my house. The time until A arrives is exponentially distributed with rate λa, and the time until B arrives is exponentially distributed with rate λb. Once
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).
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[max Xi|X1 < X2 < X3],(d) E[max Xi].
Let X, Y1,..., Yn be independent exponential random variables; X having rateλ, and Yi having rate μ. Let Aj be the event that the jth smallest of these n + 1 random variables is one of the Yi .

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