New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
business
probability and stochastic modeling
An Introduction To Stochastic Modeling 4th Edition Mark A. Pinsky, Samuel Karlin - Solutions
5.4.10 Compare and contrast the example immediately following Theorem 5.7, the shot noise process of Section 5.4.1, and the model of Problem 4.8. Can you formulate a general process of which these three examples are special cases?
5.4.9 Customers arrive at a service facility according to a Poisson process of rate customers per hour. Let N.t/ be the number of customers that have arrived up to time t, and let W1;W2; : : : be the successive arrival times of the customers.Determine the expected value of the product of the
5.4.8 Electrical pulses with independent and identically distributed random amplitudes1; 2; : : : arrive at a detector at random times W1;W2; : : : according to a Poisson process of rate . The detector output k.t/ for the kth pulse at time t isThat is, the amplitude impressed on the detector when
5.4.7 Let W1;W2; : : : be the event times in a Poisson process fX.t/I t 0g of rate , and let f .w/ be an arbitrary function. Verify that X(1) Ef(Wi) i= 0 f(w)dw.
5.4.6 Customers arrive at a service facility according to a Poisson process of rate customers/hour. Let X.t/ be the number of customers that have arrived up to time t. Let W1;W2; : : : be the successive arrival times of the customers.(a) Determine the conditional mean E[W1jX.t/ D 2].(b) Determine
5.4.5 Let W1;W2; : : : be the waiting times in a Poisson process fN.t/I t 0g of rate. Determine the limiting distribution of W1, under the condition that N.t/ D n as n!1and t!1in such a way that n=t D > 0.
5.4.4 Let W1;W2; : : : be the waiting times in a Poisson process fX.t/I t 0g of rate . Independent of the process, let Z1;Z2; : : : be independent and identically distributed random variables with common probability density function f .x/;0 < x zg, where Z D minfW1 CZ1;W2 CZ2; : : :g:
5.4.3 Let W1;W2; : : : be the waiting times in a Poisson process fX.t/I t 0g of rate . Under the condition that X.1/ D 3, determine the joint distribution of U D W1=W2 and V D .1????W3/=.1????W2/.
5.4.2 Let fN.t/I t 0g be a Poisson process of rate , representing the arrival process of customers entering a store. Each customer spends a duration in the store that is a random variable with cumulative distribution function G. The customer durations are independent of each other and of the
5.4.1 Let W1;W2; : : : be the event times in a Poisson process fX.t/I t 0g of rate .Suppose it is known that X.1/ D n. For k < n, what is the conditional density function of W1; : : : ;Wk????1;WkC1; : : : ;Wn, given that Wk D w?
5.4.5 Customers arrive at a certain facility according to a Poisson process of rate .Suppose that it is known that five customers arrived in the first hour. Each customer spends a time in the store that is a random variable, exponentially distributed with parameter and independent of the other
5.4.4 Customers arrive at a service facility according to a Poisson process of intensity. The service times Y1;Y2; : : : of the arriving customers are independent random variables having the common probability distribution function G.y/ D PrfYk yg. Assume that there is no limit to the number of
5.4.3 Customers arrive at a certain facility according to a Poisson process of rate .Suppose that it is known that five customers arrived in the first hour. Determine the mean total waiting time E[W1 CW2 C CW5].
5.4.2 Let fX.t/I t 0g be a Poisson process of rate . Suppose it is known that X.1/ D 2. Determine the mean of W1W2, the product of the first two arrival times.
5.4.1 Let fX.t/I t 0g be a Poisson process of rate . Suppose it is known that X.1/ D n. For n D 1;2; : : : ; determine the mean of the first arrival time W1.
5.3.10 Let fWng be the sequence of waiting times in a Poisson process of intensity D 1. Show that Xn D 2n expf????Wng defines a nonnegative martingale.
5.3.9 The following calculations arise in certain highly simplified models of learning processes. Let X1.t/ and X2.t/ be independent Poisson processes having parameters 1 and 2, respectively.(a) What is the probability that X1.t/ D 1 before X2.t/ D 1?(b) What is the probability that X1.t/ D 2
5.3.8 Consider a Poisson process with parameter . Given that X.t/ D n events occur in time t, find the density function for Wr, the time of occurrence of the rth event. Assume that r n.
5.3.7 A critical component on a submarine has an operating lifetime that is exponentially distributed with mean 0.50 years. As soon as a component fails, it is replaced by a new one having statistically identical properties. What is the smallest number of spare components that the submarine should
5.3.6 Customers arrive at a holding facility at random according to a Poisson process having rate . The facility processes in batches of size Q. That is, the first Q????1 customers wait until the arrival of the Qth customer. Then, all are passed simultaneously, and the process repeats. Service
5.3.5 Let X.t/ be a Poisson process with parameter . Independently, let T be a random variable with the exponential densityDetermine the probability mass function for X.T/.Hint: Use the law of total probability and Chapter 1, (1.54). Alternatively, use the results of Chapter 1, Section 1.5.2.
5.3.4 The joint probability density function for the waiting times W1 and W2 is given byDetermine the marginal density functions for W1 and W2, and check your work by comparison with Theorem 5.4. f(w, w2) = exp(-w2} for 0
5.3.3 The joint probability density function for the waiting times W1 and W2 is given byChange variables according toand determine the joint distribution of the first two sojourn times. Compare with Theorem 5.5. f(w1, w2) = exp(-2w2) for 0
5.3.2 The joint probability density function for the waiting times W1 and W2 is given byDetermine the conditional probability density function for W1, given that W2 D w2. How does this result differ from that in Theorem 5.6 when n D 2 and k D 1? f(w1, w2) exp(-2w2) for 0
5.3.1 Let X.t/ be a Poisson process of rate . Validate the identity fW1 > w1;W2 > w2g if and only if fX.w1/ D 0;X.w2/????X.w1/ D 0 or 1g :Use this to determine the joint upper tail probabilityFinally, differentiate twice to obtain the joint density function Pr{W> w, W2>w2} =
5.3.9 Let X.t/ be a Poisson process of rate . Determine the cumulative distribution function of the gamma density as a sum of Poisson probabilities by first verifying and then using the identity Wr t if and only if X.t/ r.
5.3.8 Customers arrive at a service facility according to a Poisson process of rate D 5 per hour. Given that 12 customers arrived during the first two hours of service, what is the conditional probability that 5 customers arrived during the first hour?
5.3.7 Customers arrive at a service facility according to a Poisson process of rate customers/hour. Let X.t/ be the number of customers that have arrived up to time t. Let W1;W2; : : : be the successive arrival times of the customers. Determine the conditional mean E[W5jX.t/ D 3].
5.3.6 For i D 1; : : : ;n, let fXi.t/I t 0g be independent Poisson processes, each with the same parameter . Find the distribution of the first time that at least one event has occurred in every process.
5.3.5 Let X.t/ be a Poisson process of rate per hour. Find the conditional probability that there were m events in the first t hours, given that there were n events in the first T hours. Assume 0 m n and 0 < t < T.
5.3.4 Let X.t/ be a Poisson process of rate D 3 per hour. Find the conditional probability that there were two events in the first hour, given that there were five events in the first 3 h.
5.3.3 Customers enter a store according to a Poisson process of rate D 6 per hour.Suppose it is known that only a single customer entered during the first hour.What is the conditional probability that this person entered during the first 15 min?
5.3.2 A radioactive source emits particles according to a Poisson process of rate D 2 particles per minute.(a) What is the probability that the first particle appears some time after 3 min but before 5 min?(b) What is the probability that exactly one particle is emitted in the interval from 3 to
5.3.1 A radioactive source emits particles according to a Poisson process of rate D 2 particles per minute. What is the probability that the first particle appears after 3 min?
5.2.12 Computer Challenge Most computers have available a routine for simulating a sequence U0;U1; : : : of independent random variables, each uniformly distributed on the interval .0;1/. Plot, say, 10,000 pairs .U2n, U2nC1/ on the unit square. Does the plot look like what you would expect? Repeat
5.2.11 Let X and Y be jointly distributed random variables and B an arbitrary set. Fill in the details that justify the inequality jPrfX in Bg????PrfY in Bgj PrfX 6D Yg.Hint: Begin with {X in B} = {X in B and Y in B) or (X in B and Y not in B} B) CY in B) or (XY).
5.2.10 Review the proof of Theorem 5.3 in Section 5.2.2 and establish the stronger resultfor any set of nonnegative integers I. n |Pr{S, in I) - Pr{X() in I}|P k=1
5.2.9 Using (5.6), evaluate the exact probabilities for Sn and the Poisson approximation and error bound in (5.7) when n D 4 and p1 D 0:1, p2 D 0:1, p3 D 0:1, and p4 D 0:2.
5.2.8 Using (5.6), evaluate the exact probabilities for Sn and the Poisson approximation and error bound in (5.7) when n D 4 and p1 D 0:1, p2 D 0:2, p3 D 0:3, and p4 D 0:4.
5.2.7 N bacteria are spread independently with uniform distribution on a microscope slide of area A. An arbitrary region having area a is selected for observation.Determine the probability of k bacteria within the region of areaa. Show that as N !1and a!0 such that .a=A/N !c.0 < c
5.2.6 Certain computer coding systems use randomization to assign memory storage locations to account numbers. Suppose that N D M different accounts are to be randomly located among M storage locations. Let Xi be the number of accounts assigned to the ith location. If the accounts are distributed
5.2.5 Suppose that N points are uniformly distributed over the surface of a circular disk of radius r. Determine the probability distribution for the number of points within a distance of one of the origin as N !1; r!1; N=????r2D .
5.2.4 Suppose that N points are uniformly distributed over the interval [0;N/. Determine the probability distribution for the number of points in the interval [0;1/as N !1.
5.2.3 Suppose that N pairs of socks are sent to a laundry, where they are washed and thoroughly mixed up to create a mass of unmatched socks. Then, n socks are drawn at random without replacement from the pile. Let A be the event that no pair is among the n socks so selected. Show thatUse the
5.2.2 Suppose that 100 tags, numbered 1;2; : : : ; 100, are placed into an urn, and 10 tags are drawn successively, with replacement. Let A be the event that no tag is drawn twice. Show thatUse the approximationto getInterpret this in terms of the law of rare events. Pr{A} = = (1-10) (1-170) (1 -
5.2.1 Let X.n;p/ have a binomial distribution with parameters n and p. Let n!1 and p!0 in such a way that np D . Show that lim Pr{X(n,p) = 0} = e^ and Pr{X(n,p)=k+1} lim 11-00 Pr{X(n,p)=k} k+1 for k = 0, 1, ....
5.2.4 Suppose that a book of 600 pages contains a total of 240 typographical errors.Develop a Poisson approximation for the probability that three particular successive pages are error-free.
5.2.3 A large number of distinct pairs of socks are in a drawer, all mixed up. A small number of individual socks are removed. Explain in general terms why it might be plausible to assume that the number of pairs among the socks removed might follow a Poisson distribution.
5.2.2 Explain in general terms why it might be plausible to assume that the following random variables follow a Poisson distribution:(a) The number of customers that enter a store in a fixed time period.(b) The number of customers that enter a store and buy something in a fixed time period.(c) The
5.2.1 Determine numerical values to three decimal places for PrfX D kg; k D 0;1; 2, when(a) X has a binomial distribution with parameters n D 20 and p D 0:06.(b) X has a binomial distribution with parameters n D 40 and p D 0:03.(c) X has a Poisson distribution with parameter D 1:2.
5.1.12 Consider the mixed Poisson process of Section 5.1.4, and suppose that the mixing parameter 2 has the exponential density f ./ D e???? for > 0.(a) Show that equation (5.3) becomes(b) Show thatso that X0.t/ and the increment X0.tCs/????X0.t/ are not independent random variables, in
5.1.11 Assume that a device fails when a cumulative effect of k shocks occurs. If the shocks happen according to a Poisson process of parameter , what is the density function for the life T of the device?
5.1.10 Customers arrive at a facility at random according to a Poisson process of rate . There is a waiting time cost of c per customer per unit time. The customers gather at the facility and are processed or dispatched in groups at fixed times T;2T;3T; : : : : There is a dispatch cost of K. The
5.1.9 Arrivals of passengers at a bus stop form a Poisson process X.t/ with rate D 2 per unit time. Assume that a bus departed at time t D 0 leaving no customers behind. Let T denote the arrival time of the next bus. Then, the number of passengers present when it arrives is X.T/. Suppose that the
5.1.8 Find the probability PrfX.t/ D 1;3;5; : : :g that a Poisson process having rate is odd.
5.1.7 Shocks occur to a system according to a Poisson process of rate . Suppose that the system survives each shock with probability , independently of other shocks, so that its probability of surviving k shocks is k. What is the probability that the system is surviving at time t?
5.1.6 Let fX.t/I t 0g be a Poisson process of rate . For s; t > 0, determine the conditional distribution of X.t/, given that X.tCs/ D n.
5.1.5 For each value of h > 0, let X.h/ have a Poisson distribution with parameterh. Let pk.h/ D PrfX.h/ D kg for k D 0;1; : : : : Verify thatHere o.h/ stands for any remainder term of order less than h as h!0. 1-po(h) lim h0 , or po(h) = 1-h+o(h); h lim Pi(h) h0 h lim P2(h) h0 h =0, or p2(h) =
5.1.4 (Continuation) Let X and Y be independent random variables, Poisson distributed with parameters and , respectively. Show that the generating function of their sum N D X CY is given byHint: Verify and use the fact that the generating function of a sum of independent random variables is the
5.1.3 The generating function of a probability mass function pk D PrfX D kg, for k D 0;1; : : : ; is defined byShow that the generating function for a Poisson random variable X with mean is given by 8x(s) =E[s]=Pks for |s|
5.1.2 Suppose that minor defects are distributed over the length of a cable as a Poisson process with rate , and that, independently, major defects are distributed over the cable according to a Poisson process of rate . Let X.t/ be the number of defects, either major or minor, in the cable up to
5.1.1 Let 1; 2; : : : be independent random variables, each having an exponential distribution with parameter . Define a new random variable X as follows: If1 > 1, then X D 0; if 1 1 but 1 C2 > 1, then set X D 1; in general, set X D k ifShow that X has a Poisson distribution with parameter .
5.1.9 Let fX.t/I t 0g be a Poisson process having rate parameter D 2. Determine the following expectations:(a) E[X.2/].(b) EfX.1/g2.(c) E[X.1/X.2/].
5.1.8 Let fX.t/I t 0g be a Poisson process having rate parameter D 2. Determine the numerical values to two decimal places for the following probabilities:(a) PrfX.1/ 2g.(b) PrfX.1/ D 1 and X.2/ D 3g.(c) PrfX.1/ 2jX.1/ 1g.
5.1.7 Suppose that customers arrive at a facility according to a Poisson process having rate D 2. Let X.t/ be the number of customers that have arrived up to time t.Determine the following probabilities and conditional probabilities:(a) PrfX.1/ D 2g.(b) PrfX.1/ D 2 and X.3/ D 6g.(c) PrfX.1/ D
5.1.6 Messages arrive at a telegraph office as a Poisson process with mean rate of 3 messages per hour.(a) What is the probability that no messages arrive during the morning hours 8:00 a.m. to noon?(b) What is the distribution of the time at which the first afternoon message arrives?
5.1.5 Suppose that a random variable X is distributed according to a Poisson distribution with parameter . The parameter is itself a random variable, exponentially distributed with density f .x/ D e????x for x 0. Find the probability mass function for X.
5.1.4 Customers arrive at a service facility according to a Poisson process of rate customer/hour. Let X.t/ be the number of customers that have arrived up to time t.(a) What is PrfX.t/ D kg for k D 0;1; : : :?(b) Consider fixed times 0 < s < t. Determine the conditional probability PrfX.t/ D
5.1.3 Let X and Y be independent Poisson distributed random variables with parameters and , respectively. Determine the conditional distribution of X, given that N D X CY D n.
5.1.2 Let pk D PrfX D kg be the probability mass function corresponding to a Poisson distribution with parameter . Verify that p0 D expf????g, and that pk may be computed recursively by pk D .=k/pk????1.
5.1.1 Defects occur along the length of a filament at a rate of D 2 per foot.(a) Calculate the probability that there are no defects in the first foot of the filament.(b) Calculate the conditional probability that there are no defects in the second foot of the filament, given that the first foot
4.5.1 Given the transition matrixdetermine the limits, as 0 2 34 000 314 IT P = 1 12 12 20 0 30 0 000 1 23 - 13: 0 0 0 410000
4.4.5 Let P be the transition probability matrix of a finite-state regular Markov chain.Let MD kmijk be the matrix of mean return times.(a) Use a first step argument to establish that(b) Multiply both sides of the preceding by i and sum to obtainSimplify this to show (see equation (4.26))
4.4.4 Let fi : i D 1;2; : : :g be a probability distribution, and consider the Markov chain whose transition probability matrix isWhat condition on the probability distribution fi : i D 1;2; : : :g is necessary and sufficient in order that a limiting distribution exist, and what is this limiting
4.4.3 Consider a random walk Markov chain on state 0;1; : : : ;N with transition probability matrixwhere pi Cqi D 1;pi > 0;qi > 0 for all i.The transition probabilities from state 0 and N “reflect” the process back into state 1;2; : : : ;N ????1. Determine the limiting distribution. 0 1
4.4.1 Consider the Markov chain on f0;1g whose transition probability matrix is(a) Verify that .0;1/ D .=. C/;=. C// is a stationary distribution.(b) Show that the first return distribution to state 0 is given by f .1/00 D .1????/and f .n/00 D .1????????2 for n D 2;3; : : : :(c) Calculate the mean
4.4.2 Consider the Markov chain whose transition probability matrix is given by(a) Determine the limiting probability 0 that the process is in state 0.(b) By pretending that state 0 is absorbing, use a first step analysis (Chapter 3, Section 3.4) and calculate the mean time m10 for the process to
4.3.3 Recall the first return distribution (Section 4.3.3), f = Pr{X1i, X2 j....Xn-1i,X=iXoi) for n = 1, 2,...,
4.3.1 A two-state Markov chain has the transition probability matrix(a) Determine the first return distribution(b) Verify equation (4.16) when i D 0. (Refer to Chapter 3, (4.40).) P = 0 1 01-a a b 1-b
4.3.3 A Markov chain on states f0;1;2;3;4;5g has transition probability matrixFind all communicating classes; which classes are transient and which are recurrent? 13 0 0 23 IT (a) 00 0 0 0 34 0 0 0 0 0 0 13 45 0 0 23 1416 9411 0 0 91 91 1514 16 0 IT 16 0 Ol 0 0 0 0 1470 1348 0 0 0 0 38140 18140 0
4.3.1 A Markov chain has a transition probability matrixFind the equivalence classes. For which integers n D 1;2; : : : ; 20, is it true thatWhat is the period of the Markov chain?Hint: One need not compute the actual probabilities. See Section 4.1.1. 0 1 2 3 4 0 0 1 0 0 0 100 100 2 0 0 0 1 0 3
4.2.8 An airline reservation system has a single computer, which breaks down on any given day with probability p. It takes 2 days to restore a failed computer to normal service. Form a Markov chain by taking as states the pairs .x; y/, where x is the number of machines in operating condition at the
4.2.7 Customers arrive for service and take their place in a waiting line. There is a single service facility, and a customer undergoing service at the beginning of a period will complete service and depart at the end of the period with probability and will continue service into the next period
4.2.6 Consider a computer system that fails on a given day with probability p and remains “up” with probability q D 1????p. Suppose the repair time is a random variable N having the probability mass function p.k/ D .1????/k????1 for k D 1;2; : : : ; where 0 and D 1????. Determine the long run
4.2.5 Suppose that the weather on any day depends on the weather conditions during the previous 2 days. We form a Markov chain with the following states:State .S;S/ if it was sunny both today and yesterday, State .S;C/ if it was sunny yesterday but cloudy today, State .C;S/ if it was cloudy
4.2.4 A component of a computer has an active life, measured in discrete units, that is a random variable , whereSuppose that one starts with a fresh component, and each component is replaced by a new component upon failure. Let Xn be the remaining life of the component in service at the end of
4.2.3 Suppose that a production process changes state according to a Markov process whose transition probability matrix is given by(a) Determine the limiting distribution for the process.(b) Suppose that states 0 and 1 are “In-Control,” while states 2 and 3 are deemed “Out-of-Control.” In
4.2.2 A system consists of two components operating in parallel: The system functions if at least one of the components is operating. In any single period, if both components are operating at the beginning of the period, then each will fail, independently, during the period with probability . When
4.2.1 Consider a discrete-time periodic review inventory model (see Chapter 3, Section 3.3.1), and let n be the total demand in period n. Let Xn be the inventory quantity on hand at the end of period n. Instead of following an .s;S/ policy, a.q;Q/ policy will be used: If the stock level at the end
4.2.8 At the end of a month, a large retail store classifies each receivable account according to 0: Current 1: 30–60 days overdue 2: 60–90 days overdue 3: Over 90 days Each such account moves from state to state according to a Markov chain with transition probability matrixIn the long run,
4.2.7 Consider a machine whose condition at any time can be observed and classified as being in one of the following three states:State 1: Good operating order State 2: Deteriorated operating order State 3: In repair We observe the condition of the machine at the end of each period in a sequence of
4.2.6 A component of a computer has an active life, measured in discrete units, that is a random variable T whereSuppose one starts with a fresh component, and each component is replaced by a new component upon failure. Determine the long run probability that a failure occurs in a given period.
4.2.5 From purchase to purchase, a particular customer switches brands among products A;B, and C according to a Markov chain whose transition probability matrix isIn the long run, what fraction of time does this customer purchase brand A? A B C A 0.6 0.2 0.2|| P B 0.1 0.7 0.2 P=B C 0.1 0.1 0.8
4.2.4 Section 4.2.2 determined the availability R of a certain computer system to bewhere p is the computer failure probability on a single day. Compute and compare R1 and R2 for p D 0:01;0:02;0:05, and 0:10. R for one repair facility, 1+p 1+p R = for two repair facilities, 1+p+p
4.2.3 Determine the average fraction inspected, AFI, and the average outgoing quality, AOQ, of Section 4.2.3 for p D 0;0:05;0:10;0:15; : : : ;0:50 when(a) r D 10 and i D 5.(b) r D 5 and i D 10.
4.2.2 In the reliability example of Section 4.2.2, what fraction of time is the repair facility idle? When a second repair facility is added, what fraction of time is each facility idle?
4.2.1 On a southern Pacific island, a sunny day is followed by another sunny day with probability 0.9, whereas a rainy day is followed by another rainy day with probability 0.2. Supposing that there are only sunny or rainy days, in the long run on what fraction of days is it sunny?
4.1.13 A Markov chain has the transition probability matrixAfter a long period of time, you observe the chain and see that it is in state 1. What is the conditional probability that the previous state was state 2? That is, find 012 0 0.4 0.4 0.2 P 1 0.6 0.2 0.2 2 0.4 0.2 0.4
4.1.12 Let P be the transition probability matrix of a finite-state regular Markov chain, and let 5 be the matrix whose rows are the stationary distribution . Define Q D P????5.(a) Show that Pn D 5CQn.(b) Whenobtain an explicit expression for Qn and then for Pn. 222 1214 0 P
4.1.11 Suppose that a production process changes state according to a Markov process whose transition probability matrix is given byIt is known that 1 D 119 379 D 0:3140 and 2 D 81 379 D 0:2137.(a) Determine the limiting probabilities 0 and 3.(b) Suppose that states 0 and 1 are “In-Control”
Showing 4200 - 4300
of 6914
First
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Last
Step by Step Answers
Get 50% OFF
Claim Now
