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theory of probability

Questions and Answers of Theory Of Probability

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 {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 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.
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).
In Exercise 40 what is the
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
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
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 )).
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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.
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
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 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
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
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
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
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
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
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
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
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
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
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
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
For the model of Example 5.27, find the mean and variance of the number of customers served in a busy period.
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
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 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
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
(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 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
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 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
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
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
Determine Cov[X(t), X(t + s)]when {X(t), t 0} is a compound Poisson process.
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
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
In Exercise 89 show that X1 and X2 both have exponential distributions.
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
Prove Equation (5.22).
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)+i< j
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
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].
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.
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.
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)
Let X be the time between the first and the second event of a Hawkes process with mark distribution F. Find P(X > t).
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
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
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
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
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
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 expected time to
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
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 rateμ. Set up the Kolmogorov backward
The birth and death process with parameters λn = 0 and μn = μ, n > 0 is called a pure death process. Find Pi j(t)
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 other.
Consider a Yule process starting with a single individual—that is, suppose X(0) = 1. Let Ti denote the time it takes the process to go from a population of size i to one of size i + 1.(a) Argue
Each individual in a biological population is assumed to give birth at an exponential rate λ, and to die at an exponential rate μ. In addition, there is an exponential rate of increase θ due to
A small barbershop, operated by a single barber, has room for at most two customers. Potential customers arrive at a Poisson rate of three per hour, and the successive service times are independent
Potential customers arrive at a full-service, one-pump gas station at a Poisson rate of 20 cars per hour. However, customers will only enter the station for gas if there are no more than two cars
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
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
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
After being repaired, a machine functions for an exponential time with rate λand then fails. Upon failure, a repair process begins. The repair process proceeds sequentially through k distinct
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 λi,i = 1, 2. When machine i fails, it requires an
There are two machines, one of which is used as a spare. A working machine will function for an exponential time with rate λ and will then fail. Upon failure, it is immediately replaced by the other
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
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
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
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
Customers arrive at a service station, manned by a single server who serves at an exponential rate μ1, at a Poisson rate λ. After completion of service the customer?
Consider an ergodic M/M/s 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
In the M/M/s 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
If {X(t)} and {Y (t)} are independent continuous-time Markov chains, both of which are time reversible, show that the process {X(t), Y (t)} is also a time reversible Markov chain.
Consider a set of n machines and a single repair facility to service these machines.Suppose that when machine i,i = 1,..., n, fails it requires an exponentially distributed amount of work with rate
Consider a graph with nodes 1, 2,..., n and the n 2arcs (i, j),i = j,i, j, =1,..., n. (See Section 3.6.2 for appropriate definitions.) Suppose that a particle moves along this graph as follows:
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, j = i, with probability 1/(r − 1). If the server he
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
Consider two M/M/1 queues with respective parameters λi, μi,i = 1, 2. Suppose they share a common waiting room that can hold at most three customers.That is, whenever an arrival finds her server
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
Consider a time reversible continuous-time Markov chain having infinitesimal transition rates qi j and limiting probabilities {Pi}. Let A denote a set of states for this chain, and consider a new
Consider a system of n components such that the working times of component i, i = 1,..., n, are exponentially distributed with rate λi . When a component fails, however, the repair rate of component
A hospital accepts k different types of patients, where type i patients arrive according to a Poisson proccess with rate λi , with these k Poisson processes being independent. Type i patients spend
Consider an n server system where the service times of server i are exponentially distributed with rate μi, i = 1,..., n. Suppose customers arrive in accordance with a Poisson process with rate λ,

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