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theory of probability

Questions and Answers of Theory Of Probability

Let X and Y be independent exponential random variables with respective ratesλ and μ. Let M = min(X, Y ). Find(a) E[M X|M = X],(b) E[M X|M = Y ],(c) Cov(X, M).
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 the
If X and Y are independent exponential random variables with respective rates λand μ, what is the conditional distribution of X given that X < Y ?
If X1 and X2 are independent nonnegative continuous random variables, show that P{X1 < X2| min(X1, X2) = t} = r1(t)r1(t) + r2(t)where ri(t) is the failure rate function of Xi .
In Example 5.3 if server i serves at an exponential rate λi,i = 1, 2, show that P{Smith is not last} = λ1λ1 + λ2 2+λ2λ1 + λ2 2
If X is exponential with rate λ, show that Y = [X]+1 is geometric with parameter p = 1 − e−λ, where [x] is the largest integer less than or equal to x.
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
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) E[X2|X >
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 are
The time T required to repair a machine is an exponentially distributed random variable with mean 1 2 (hours).(a) What is the probability that a repair time exceeds 1 2 hour?(b) What is the
Verify Equation (9.36).
Prove the combinatorial identity?
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.
Show that the variance of the lifetime of a k-out-of-n system of components, each of whose lifetimes is exponential with mean θ, is given byθ 2n i=k 1i 2
Show that the mean lifetime of a parallel system of two components is 1μ1 + μ2+ μ1(μ1 + μ2)μ2+ μ2(μ1 + μ2)μ1 when the first component is exponentially distributed with mean 1/μ1 and the
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.
We say that ζ is a p-percentile of the distribution F if F(ζ ) = p. Show that if ζis a p-percentile of the IFRA distribution F, then F¯(x) e−θ x , x ζF¯(x) e−θ x , x ζ
Show that if F is IFR, then it is also IFRA, and show by counterexample that the reverse is not true.
Show that if each (independent) component of a series system has an IFR distribution, then the system lifetime is itself IFR by(a) showing thatλF (t) =iλi(t)where λF (t) is the failure rate
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 F¯t(a) = P{Xt > a}In words, F¯t(a) is the probability that a t-year-old
Consider the following four structures:(i) See Figure 9.23:
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 distribution
Consider a structure in which the minimal path sets are {1, 2, 3} and {3, 4, 5}.(a) What are the minimal cut sets?
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 Figure 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 minimal path sets and the reliability function for the structure in Figure 9.22.Figure 9.22
Give the reliability function of the structure of Exercise 8.
Let ti denote the time of failure of the ith component; let τφ(t) denote the time to failure of the system φ as a function of the vector t = (t1,..., tn). Show that max 1 js min i∈Aj ti =
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.
Give the minimal path sets and the minimal cut sets for the structure given by Figure 9.21
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.
Find the minimal path and minimal cut sets for:(a) See Figure 9.19:
Write the structure function corresponding to the following:(a) See Figure 9.16:Figure 9.16(b) See Figure 9.17:
For any structure function, 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 a
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
Consider a system where the interarrival times have an arbitrary distribution F, and there is a single server whose service distribution is G. Let Dn denote the amount of time the nth customer spends
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 = .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 24.
Consider the priority queueing model of Section 8.6.2 but now suppose that if a type 2 customer is being served when a type 1 arrives then the type 2 customer is bumped out of service. This is called
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] and greater than 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
Consider an M/G/1 system in which the first customer in a busy period has the service distribution G1 and all others have distribution G2. Let C denote the number of customers in a busy period, and
For the M/G/1 queue, let Xn denote the number in the system left behind by the nth departure.(a) If?
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
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?
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
For the tandem queue model verify that Pn,m = (λ/μ1)n(1 − λ/μ1)(λ/μ2)m(1 − λ/μ2)satisfies the balance Equation (8.15).
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
Let D denote the time between successive departures in a stationary M/M/1 queue with λ
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
Poisson (λ) arrivals join a queue in front of two parallel servers A and B, having exponential service rates μA and μB (see Figure 8.4). When the system is empty, arrivals go into server A with
Reconsider Exercise 23, 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
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 20 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 20, 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 rate λ.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 rateλ1, 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,
A supermarket has two exponential checkout counters, each operating at rate μ.Arrivals are Poisson at rate λ. The counters operate in the following way:(i) One queue feeds both counters.(ii) One
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?
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 8.7 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 λ and with each server at rate μ. Can you
It follows from Exercise 4 that if, in the M/M/1 model, W∗Q is the amount of time that a customer spends waiting in queue, then W∗Q =0, with probability 1 − λ/μExp(μ − λ), with
Suppose that a customer of the M/M/1 system spends the amount of time x > 0 waiting in queue before entering service.(a) Show that, conditional on the preceding, the number of other customers that
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.”
Let h(x) = P{T i=1 Xi > x} where X1, X2,... are independent random variables having distribution function Fe and T is independent of the Xi and has probability mass function P{T = n} = ρn(1 −
Random digits, each of which is equally likely to be any of the digits 0 through 9, are observed in sequence.(a) Find the expected time until a run of 10 distinct values occurs.(b) Find the expected

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