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theory of probability

Questions and Answers of Theory Of Probability

Compute the conditional distribution of B(s) given that B(t1) = A and B(t2) =B, where 0 < t1 < s < t2.
What is the distribution of B(s) + B(t), s ≤ t?
Verify Eq. (9.36).
For the model of Section 9.7, compute for a k-out-of-n structure (i) the average up time, (ii) the average down time, and (iii) the system failure rate.
Let Xi be an exponential random variable with mean 8 + 2i, for i = 1, 2, 3.Use the results of Section 9.6.1 to obtain an upper bound on E[maxXi ], and then compare this with the exact result when the
In Section 9.6.1, show that the expected number of Xi that exceed c∗ is equal to 1.
Compute the expected system lifetime of a three-out-of-four system when the first two component lifetimes are uniform on (0, 1) and the second two are uniform on (0, 2).
Find the mean lifetime of a series system of two components when the component lifetimes are respectively uniform on (0, 1) and uniform on (0, 2). Repeat for a parallel system.
Let r(p) = r(p, p, . . . , p). Show that if r(p0) = p0, then r(p) p for p p0 r(p) p for p p0 Hint: Use Proposition 9.2.
Prove Lemma 9.3.Hint: Let x = y + δ. Note that f (t) = tα is a concave function when 0 α 1, and use the fact that for a concave function f (t + h) − f (t) is decreasing in t .
Show that if F is IFR, then it is also IFRA, and show by counterexample that the reverse is not true.
Let X denote the lifetime of an item. Suppose the item has reached the age of t .Let Xt denote its remaining life and define Ft (a) = P{Xt >a}In words, ¯ Ft (a) is the probability that a t-year-old
Let F be a continuous distribution function. For some positive α, define the distribution function G by¯ G(t) = ( ¯ F(t))αFind the relationship between λG(t) and λF (t), the respective failure
Let X1,X2, . . . , Xn denote independent and identically distributed random variables and define the order statistics X(1), . . . , X(n) by X(i) ≡ ith smallest of X1, . . . , Xn Show that if the
Consider a structure in which the minimal path sets are {1, 2, 3} and {3, 4, 5}.(a) What are the minimal cut sets?(b) If the component lifetimes are independent uniform (0, 1) random variables,
Let N be a nonnegative, integer-valued random variable. Show that P{N >0} (E[N])2 E[N2]and explain how this inequality can be used to derive additional bounds on a reliability function.Hint:E[N2] =
Compute the upper and lower bounds of r(p) using both methods for the(a) two-out-of-three system and(b) two-out-of-four system.(c) Compare these bounds with the exact reliability when(i) pi ≡
Compute upper and lower bounds of the reliability function (using Method 2)for the systems given in Exercise 4, and compare them with the exact values when pi ≡ 1 2 .
Compute the reliability function of the bridge system (see Fig. 9.11) by conditioning upon whether or not component 3 is working.
Let r(p) be the reliability function. Show that r(p) = pir(1i ,p)+ (1 −pi)r(0i ,p)
Give the reliability function of the structure of Exercise 8.
Component i is said to be relevant to the system if for some state vector x,φ(1i , x) = 1, φ(0i , x) = 0 Otherwise, it is said to be irrelevant.(a) Explain in words what it means for a component to
The minimal cut sets are {1, 2, 3}, {2, 3, 4}, and {3, 5}. What are the minimal path sets?
The minimal path sets are {1, 2, 4}, {1, 3, 5}, and {5, 6}. Give the minimal cut sets.
For any structure function φ, we define the dual structure φD byφD(x) = 1− φ(1 −x)(a) Show that the dual of a parallel (series) system is a series (parallel) system.(b) Show that the dual of
Show that(a) if φ(0, 0, . . . , 0) = 0 and φ(1, 1, . . . , 1) = 1, then minxi φ(x) maxxi(b) φ(max(x, y)) max(φ(x),φ(y))(c) φ(min(x, y)) min(φ(x),φ(y))
Prove that, for any structure function φ,φ(x) = xiφ(1i , x)+(1 −xi)φ(0i , x)where (1i , x) = (x1, . . . , xi−1, 1, xi+1, . . . , xn), (0i , x) = (x1, . . . , xi−1, 0, xi+1, . . . , xn)
Consider a model in which the interarrival times have an arbitrary distribution F, and there are k servers each having service distribution G. What condition on F and G do you think would be
Verify the formula for the distribution of W∗Q given for the G/M/k model.
In the M/M/k system,(a) what is the probability that a customer will have to wait in queue?(b) determine L and W.
In the Erlang loss system suppose the Poisson arrival rate is λ = 2, and suppose there are three servers, each of whom has a service distribution that is uniformly distributed over (0, 2). What
Verify the formula given for the Pi of the M/M/k.
In the k server Erlang loss model, suppose that λ = 1 and E[S] = 4. Find L if Pk = 0.2.
In the G/M/1 model if G is exponential with rate λ show that β = λ/μ.
Calculate explicitly (not in terms of limiting probabilities) the average time a customer spends in the system in Exercise 28.
In a two-class priority queueing model suppose that a cost of Ci per unit time is incurred for each type i customer that waits in queue, i = 1, 2. Show that type 1 customers should be given priority
In the two-class priority queueing model of Section 8.6.2, what is WQ? Show that WQ is less than it would be under FIFO if E[S1]E[S2].
Carloads of customers arrive at a single-server station in accordance with a Poisson process with rate 4 per hour. The service times are exponentially distributed with rate 20 per hour. If each
Consider a M/G/1 system with λE[S] < 1.(a) Suppose that service is about to begin at a moment when there are n customers in the system.(i) Argue that the additional time until there are only n − 1
In an M/G/1 queue,(a) what proportion of departures leave behind 0 work?(b) what is the average work in the system as seen by a departure?
Compare the M/G/1 system for first-come, first-served queue discipline with one of last-come, first-served (for instance, in which units for service are taken from the top of a stack).Would you think
Customers arrive at a single-server station in accordance with a Poisson process having rate λ. Each customer has a value. The successive values of customers are independent and come from a uniform
For open queueing networks(a) state and prove the equivalent of the arrival theorem;(b) derive an expression for the average amount of time a customer spends waiting in queues.
Explain how a Markov chain Monte Carlo simulation using the Gibbs sampler can be utilized to estimate(a) the distribution of the amount of time spent at server j on a visit.Hint: Use the arrival
Consider a closed queueing network consisting of two customers moving among two servers, and suppose that after each service completion the customer is equally likely to go to either server—that
Consider a network of three stations with a single server at each station. Customers arrive at stations 1, 2, 3 in accordance with Poisson processes having respective rates 5, 10, and 15. The service
Potential customers arrive to a single-server hair salon according to a Poisson process with rate λ. A potential customer who finds the server free enters the system; a potential customer who finds
Consider a single-server exponential system in which ordinary customers arrive at a rate λ and have service rate μ. In addition, there is a special customer who has a service rate μ1. Whenever
In a queue with unlimited waiting space, arrivals are Poisson (parameter λ)and service times are exponentially distributed (parameter μ). However, the server waits until K people are present before
Poisson (λ) arrivals join a queue in front of two parallel servers A and B, having exponential service rates μA and μB (see Fig. 8.4). When the system is empty, arrivals go into server A with
Reconsider Exercise 27, but this time suppose that a customer that is in the system when a breakdown occurs remains there while the server is being fixed.In addition, suppose that new arrivals during
Consider the M/M/1 system in which customers arrive at rate λ and the server serves at rate μ. However, suppose that in any interval of length h in which the server is busy there is a probability
Customers arrive at a single-server station in accordance with a Poisson process with rate λ. All arrivals that find the server free immediately enter service.All service times are exponentially
Suppose in Exercise 24 we want to find out the proportion of time there is a type 1 customer with server 2. In terms of the long-run probabilities given in Exercise 24, what is(a) the rate at which a
There are two types of customers. Type 1 and 2 customers arrive in accordance with independent Poisson processes with respective rate λ1 and λ2. There are two servers. A type 1 arrival will enter
The economy alternates between good and bad periods. During good times customers arrive at a certain single-server queueing system in accordance with a Poisson process with rate λ1, and during bad
Arrivals to a three-server system are according to a Poisson process with rateλ. Arrivals finding server 1 free enter service with 1. Arrivals finding 1 busy but 2 free enter service with 2.
Customers arrive at a two-server station in accordance with a Poisson process with a rate of two per hour. Arrivals finding server 1 free begin service with that server. Arrivals finding server 1
Customers arrive at a two-server system according to a Poisson process having rate λ = 5. An arrival finding server 1 free will begin service with that server. An arrival finding server 1 busy and
Consider a sequential-service system consisting of two servers, A and B. Arriving customers will enter this system only if server A is free. If a customer does enter, then he is immediately served by
Consider a queueing system having two servers and no queue. There are two types of customers. Type 1 customers arrive according to a Poisson process having rate λ1, and will enter the system if
Two customers move about among three servers. Upon completion of service at server i, the customer leaves that server and enters service at whichever of the other two servers is free. (Therefore,
Consider a 2-server system where customers arrive according to a Poisson process with rate λ, and where each arrival is sent to the server currently having the shortest queue. (If they have the same
Customers arrive to a two server system in accordance with a Poisson process with rate λ. Server 1 is the preferred server, and an arrival finding server 1 free enters service with 1; an arrival
Customers arrive to a single server system in accordance with a Poisson process with rate λ. Arrivals only enter if the server is free. Each customer is either a type 1 customer with probability p
Families arrive at a taxi stand according to a Poisson process with rate λ. An arriving family finding N other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to
A group of m customers frequents a single-server station in the following manner.When a customer arrives, he or she either enters service if the server is free or joins the queue otherwise. Upon
A group of n customers moves around among two servers. Upon completion of service, the served customer then joins the queue (or enters service if the server is free) at the other server. All service
A facility produces items according to a Poisson process with rate λ. However, it has shelf space for only k items and so it shuts down production whenever k items are present. Customers arrive at
Consider the M/M/1 queue with impatient customers model as presented in Example 8.9. Give your answers in terms of the limiting probabilities Pn,n ≥ 0.(a) What is the average amount of time that a
Show that W is smaller in an M/M/1 model having arrivals at rate λ and service at rate 2μ than it is in a two-server M/M/2 model with arrivals at rate λand with each server at rate μ. Can you
Suppose customers arrive to a two server system according to a Poisson process with rate λ, and suppose that each arrival is, independently, sent either to server 1 with probability α or to server
In the M/M/1 system, derive P0 by equating the rate at which customers arrive with the rate at which they depart.
The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of $3 per hour. Alice, who gives service at an
Machines in a factory break down at an exponential rate of six per hour. There is a single repairman who fixes machines at an exponential rate of eight per hour. The cost incurred in lost production
For the M/M/1 queue, compute(a) the expected number of arrivals during a service period and(b) the probability that no customers arrive during a service period.Hint: “Condition.”
There are n patients needing a kidney transplant. Kidneys arrive according to a Poisson process with rate λ, and each patient is independently eligible to receive an arriving kidney with probability
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
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
Events occur according to a Poisson process N(t), t ≥ 0, with rate λ. An event for which there are no other events within a time d of it is said to be isolated.That is, an event occurring at timey
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
If X1, . . . , Xn are independent exponential random variables with rate λ, find(a) P(X1
For the model of Example 5.27, find the mean and variance of the number of customers served in a busy period.
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, roughly how many lawyers would you
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
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 position
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)
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
Cars pass an intersection according to a Poisson process with rate λ. There are 4 types of cars, and each passing car is, independently, type i with probability pi , 4 i=1 pi = 1.(a) Find the
People arrive according to a Poisson process with rate λ, with each person independently being equally likely to be either a man or a woman. If a woman(man) arrives when there is at least one man
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. Sup pose 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

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