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probability and stochastic modeling

An Introduction To Stochastic Modeling 4th Edition Mark A. Pinsky, Samuel Karlin - Solutions

9.2.3 Customers arrive at a checkout station in a market according to a Poisson process of rate D 1 customer per minute. The checkout station can be operated with or without a bagger. The checkout times for customers are exponentially distributed, and with a bagger the mean checkout time is 30 s,
9.2.2 On a single graph, plot the server utilization 1????0 D and the mean queue length L D =.1????/ for the M=M=1 queue as a function of the traffic intensity D = for 0 < < 1.
9.2.1 Customers arrive at a tool crib according to a Poisson process of rate D 5 per hour. There is a single tool crib employee, and the individual service times are exponentially distributed with a mean service time of 10 min. In the long run, what is the probability that two or more workers are
9.1.1 Two dump trucks cycle between a gravel loader and a gravel unloader. Suppose that the travel times are insignificant relative to the load and unload times, which are exponentially distributed with parameters and , respectively. Model the system as a closed queueing network. Determine the
9.1.1 What design questions might be answered by modeling the following queueing systems?What might be reasonable assumptions concerning the arrival process, service distribution, and priority in these instances? The Customer (a) Arriving airplanes (b) Cars (c) Broken TVs (d) Patients (e) Fires The
8.5.4 In the Ehrenfest urn model (see Chapter 3, Section 3.3.2) for molecular diffusion through a membrane, if there are i particles in urn A, the probability that there will be iC1 after one time unit is 1????i=.2N/, and the probability of i????1 is i=.2N/, where 2N is the aggregate number of
8.5.3 Verify the option valuation formulation (8.66).Hint: Use the result of Exercise 8.4.6.
8.5.2 Let S.t/ be the position process corresponding to an Ornstein–Uhlenbeck velocity V.t/. Assume that S.0/ D V.0/ D 0. Obtain the covariance between S.t/ and V.t/.
8.5.1 Let 1; 2; : : : be independent standard normal random variables and a constant, 0 (a) Show thatComment on the comparison with (8.78).(b) Let 1Vn D Vn ????Vn????1;S1 D vCV1 C CVn, and Bn D 1 C Cn.Show thatCompare and contrast with (8.80). Vo=v and V = (1-B)Vn-1+n for n1. n
8.5.3 Let 1; 2; : : : be independent standard normal random variables and a constant, 0 a) Determine the mean value function and covariance function for fVng.(b) Let 1V D VnC1 ????Vn. Determine the conditional mean and variance of 1V, given that Vn D v. Vo=v and V = (1-B)Vn-1+ for n1.
8.5.2 The velocity of a certain particle follows an Ornstein–Uhlenbeck process with2 D 1 and D 0:2. The particle starts at rest .v D 0/ from position S.0/ D 0.What is the probability that it is more than one unit away from its origin at time t D 1. What is the probability at times t D 10 and t D
8.5.1 An Ornstein–Uhlenbeck process V.t/ has 2 D 1 and D 0:2. What is the probability that V.t/ 1 for t D 1; 10, and 100? Assume that V.0/ D 0.
8.4.10 Let t0 D 0 < t1 < t2 < be time points, and define Xn D Z.tn/exp.????rtn/, where Z.t/ is geometric Brownian motion with drift parameters r and variance parameter 2 (see the geometric Brownian motion in the Black-Scholes formula (8.53)). Show that fXng is a martingale.
8.4.9 Let be the first time that a standard Brownian motion B.t/ starting from B.0/ D x > 0 reaches zero. Let be a positive constant. Show that w(x)=E[e|B(0)=x] = e2xx Hint: Develop an appropriate differential equation by instituting an infinites- imal first step analysis according to
8.4.8 Verify the Hewlett-Packard option valuation of $6.03 stated in the text when D 1 2 ; z D $59;a D 60; r D 0:05, and D 0:35. What is the Black-Scholes valuation if D 0:30?
8.4.6 What is the probability that a geometric Brownian motion with drift parameter D 0 ever rises to more than twice its initial value? (You buy stock whose fluctuations are described by a geometric Brownian motion with D 0. What are your chances to double your money?)
8.4.5 Change a Brownian motion with drift X.t/ into an absorbed Brownian motion with drift XA.t/ by definingwhere D minft 0IX.t/ D 0g:(We suppose that X.0/ D x > 0 and that x? X^ (t) = x(t), 10. fort
8.4.4 A Brownian motion X.t/ either (1) has drift D 0, or (2) has drift D 1, where 0 < 1 are known constants. It is desired to determine which is the case by observing the process. Derive a sequential decision procedure that meets prespecified error probabilities and .Hint: Base your
8.4.3 If B0.s/;0 is a standard Brownian motion. Use this representation and the result of Problem 8.4.2 to show that for a Brownian bridge B0.t/, B) = (1+DB (1+1)
8.4.2 Show that Prom Pr max 1>0 b+B(t) 1+t ' > a} = e-2a(a-b), a>0,b
8.4.1 What is the probability that a standard Brownian motion fB.t/g ever crosses the line aCbt.a > 0;b > 0/?
8.4.6 Let be a standard normal random variable.(a) For an arbitrary constanta, show that(b) Let X be normally distributed with mean and variance 2. Show that E[-a)]=(a) a[1-(a)].
8.4.5 Suppose that the fluctuations in the price of a share of stock in a certain company are well described by a geometric Brownian motion with drift D ????0:1 and variance 2 D 4. A speculator buys a share of this stock at a price of $100 and will sell if ever the price rises to $110 (a profit)
8.4.4 A Brownian motion X.t/ either (1) has drift D C1 2 > 0, or (2) has drift D ????1 2 < 0, and it is desired to determine which is the case by observing the process for a fixed duration . If X. / > 0, then the decision will be that D C12; If X. / 0, then D ????12 will be
8.4.3 A Brownian motion fX.t/g has parameters D 0:1 and D 2. Evaluate the mean time to exit the interval .a;b] from X(0) D 0 for b D 1; 10, and 100 and a D ????b. Can you guess how this mean time varies with b for b large?
8.4.2 A Brownian motion fX.t/g has parameters D 0:1 and D 2. Evaluate the probability of exiting the interval .a;b] at the point b starting from X.0/ D 0 for b D 1; 10, and 100 and a D ????b. Why do the probabilities change when a=b is the same in all cases?
8.4.1 A Brownian motion fX.t/g has parameters D ????0:1 and D 2. What is the probability that the process is above y D 9 at time t D 4, given that it starts at x D 2:82?
8.3.8 Show that the transition densities for both reflected Brownian motion and absorbed Brownian motion satisfy the diffusion equation (8.3) in the region 0 < x
8.3.7 Let t0 D 0 < t1 < t2 < be time points, and define Xn D A.tn/, where A.t/ is absorbed Brownian motion starting from A.0/ D x. Show that fXng is a nonnegative martingale. Compare the maximal inequality (2.53) in Chapter 2 with the result in Problem 8.3.6.
8.3.6 Let M D maxfA.t/I t 0g be the largest value assumed by an absorbed Brownian motion A.t/. Show that PrfM > zjA.0/ D xg D x=z for 0 < x < z.
8.3.5 Determine the expected value for absorbed Brownian motion A.t/ at time t D 1 by integrating the transition density (8.32) according toThe answer is E[A.1/jA.0/ D x] D x. Show that E[A.t/jA.0/ D x] D x for all t > 0. E[A(1)|A(0)=x]=yp(y, 1x)dy 0 = y[(y-x) - (y+x)]dy. 0
8.3.3 Let B.t/ be a standard Brownian motion. Show that B.u/????uB.1/;0 < u < 1, is independent of B.1/.(a) Use this to show that B0.t/ D B.t/????tB.1/;0 t 1, is a Brownian bridge.(b) Use the representation in (a) to evaluate the covariance function for a Brownian bridge.
8.3.2 Let B.t/ be a standard Brownian motion process. Determine the conditional mean and variance of B.t/;0 < t < 1, given that B.1/ D b.
8.3.1 Let B1.t/ and B2.t/ be independent standard Brownian motion processes. DefineR.t/ is the radial distance to the origin in a two-dimensional Brownian motion.Determine the mean of R.t/. R(t)=B(t)+B(1), t0.
8.3.5 Is reflected Brownian motion a Gaussian process? Is absorbed Brownian motion(cf. Section 8.1.4)?
8.3.1 Show that the cumulative distribution function for reflected Brownian motion isEvaluate this probability when x D 1; y D 3, and t D 4. Pr{R() < y\R(0) = x) = 0 (**) - 0 (= -y-x y+x =*(**)+*(**)-1 -(*)-(7)
8.2.6 Use the result of Problem 8.2.5 to show that Y.t/ D M.t/????B.t/ has the same distribution as jB.t/j.
8.2.5 Show that the joint density function for M.t/ and Y.t/ D M.t/????B.t/ is given by z+y 2 fM),Y)(z, y) = z+y t
8.2.4 Use the reflection principle to obtain(M.t/ is the maximum defined in (8.19).) Differentiate with respect to x, and then with respect to z, to obtain the joint density function for M.t/ and B.t/: Pr{M(t)>z, B(t)
8.2.3 For a fixed t > 0, show that M.t/ and jB.t/j have the same marginal probability distribution, whenceFor 0 jB.t/j/? fM(1)(2) = 2 () for z> 0. (Here M(t)=maxout B(u).) Show that E[M(t)]=21/.
8.2.2 Find the conditional probability that a standard Brownian motion is not zero in the interval .0;b] given that it is not zero in the interval .0;a], where 0 < a < b.Hint: Let t!0 in the result of Problem 8.2.1.
8.2.1 Find the conditional probability that a standard Brownian motion is not zero in the interval .t; tCb] given that it is not zero in the interval .t; tCa], where 0 < a < b and t > 0.
8.2.6 Let 1 be the smallest zero of a standard Brownian motion that exceeds b > 0.Show that Prf1 < tg D 2arccos pb=t:
8.2.5 Let 0 be the largest zero of a standard Brownian motion not exceeding a > 0.That is, 0 D maxfu 0IB.u/ D 0 and u ag. Show that Prf0 < tg D 2arcsin pt=a:
8.2.4 Consider the simple random walk Sn D 1 C Cn; S0 D 0;in which the summands are independent with Prf D 1g D 12. Let Mn D max0kn Sk. Use a reflection argument to show that PrfMn ag D 2PrfSn > agCPrfSn D ag; a > 0:
8.2.2 Show that arctan ps=t D arccos pt=.sCt/:
8.2.1 Let fB.t/I t 0g be a standard Brownian motion, with B.0/ D 0, and let M.t/ D maxfB.u/I 0 u tg.(a) Evaluate PrfM.4/ 2g.(b) Find the number c for which PrfM.9/ > cg D 0:10.
8.1.6 Manufacturers of crunchy munchies such as cheese crisps use compression testing machines to gauge product quality. The crisp, or whatever, is placed between opposing plates, which then move together. As the crisp is crunched, the force is measured as a function of the distance that the plates
8.1.5 Consider the simple random walkin which the summands are independent with Prf D 1g D 1 2 . We are going to stop this random walk when it first drops a units below its maximum to date.Accordingly, let(a) Use a first step analysis to show thatIdentify the distribution of M .(c) Let B.t/ be
8.1.4 Let 1; : : : ;n be real constants. Argue thatis normally distributed with mean zero and variance n (1) i=1
8.1.3 For a positive constant , show thatHow does this behave when t is large .t!1/? How does it behave when t is small .t 0/? |B(t)\ Pr > e} = 2(10(1)). r { \B (0) | > } = t
8.1.2 Evaluate EeB.t/for an arbitrary constant and standard Brownian motion B.t/.
8.1.1 Consider the simple random walk Sn D 1 C Cn; S0 D 0;in which the summands are independent with Prf D 1g D 1 2 . In Chapter 3, Section 3.5.3, we showed that the mean time for the random walk to first reach????a < 0 or b > 0 is ab. Use this together with the invariance principle to show
7.6.4 Determine the long run population growth rate for a population whose individual net maternity function is m0 D m1 D 0 and m2 D m3 D D a > 0. Compare this with the population growth rate when m2 Da, and mk D 0 for k 6D 2.
7.6.3 Determine the long run population growth rate for a population whose individual net maternity function is m2 D m3 D 2, and mk D 0 otherwise. Why does delaying the age at which offspring are first produced cause a reduction in the population growth rate? (The population growth rate when m1 D
7.6.2 Marlene has a fair die with the usual six sides. She throws the die and records the number. She throws the die again and adds the second number to the first. She repeats this until the cumulative sum of all the tosses first exceeds a prescribed number n. (a) When n D 10, what is the
7.6.1 Suppose the lifetimes X1;X2; : : : have the geometric distributionwhere 0 (a) Determine un for n D 1;2; : : : :(b) Determine the distribution of excess life n by using Lemma 7.1 and (7.36). Pr{X =k) a(1-a)-1 for k = 1, 2,...,
7.6.3 Using the data of Exercises 7.6.1 and 7.6.2, determine(a) limn!1un.(b) limn!1vn.
7.6.2 (Continuation of Exercise 7.6.1)(a) Solve for un for n D 0;1; : : : ;10 in the renewal equationwhere 0 D 1; 1 D 2 D D 0, and fpkg is as defined in Exercise 7.6.1.(b) Verify that the solution vn in Exercise 7.6.1 and un are related according to vn D 6n kD0bkun????k. n == un=8+Pkn-k for
7.6.1 Solve for vn for n D 0;1; : : : ;10 in the renewal equation Vn=bn+PkVn-k for n= 0,1,...., k=0
7.5.4 A lazy professor has a ceiling fixture in his office that contains two lightbulbs.To replace a bulb, the professor must fetch a ladder, and being lazy, when a single bulb fails, he waits until the second bulb fails before replacing them both. Assume that the length of life of the bulbs are
7.5.3 At the beginning of each period, customers arrive at a taxi stand at times of a renewal process with distribution law F.x/. Assume an unlimited supply of cabs, such as might occur at an airport. Suppose that each customer pays a random fee at the stand following the distribution law G.x/, for
7.5.2 The random lifetime X of an item has a distribution function F.x/. What is the mean total life E[XjX > x] of an item of age x?
7.5.1 A certain type component has two states: 0 D OFF and 1 D OPERATING. In state 0, the process remains there a random length of time, which is exponentially distributed with parameter , and then moves to state 1. The time in state 1 is exponentially distributed with parameter , after which the
7.5.3 Consider a lightbulb whose life is a continuous random variable X with probability density function f .x/, for x > 0. Assuming that one starts with a fresh bulb and that each failed bulb is immediately replaced by a new one, let M.t/ D E[N.t/] be the expected number of renewals up to time
7.5.2 The weather in a certain locale consists of alternating wet and dry spells. Suppose that the number of days in each rainy spell is Poisson distributed with parameter 2, and that a dry spell follows a geometric distribution with a mean of 7 days. Assume that the successive durations of rainy
7.5.1 Jobs arrive at a certain service system according to a Poisson process of rate .The server will accept an arriving customer only if it is idle at the time of arrival.Potential customers arriving when the system is busy are lost. Suppose that the service times are independent random variables
7.4.5 A Markov chain X0;X1;X2; : : : has the transition probability matrixA sojourn in a state is an uninterrupted sequence of consecutive visits to that state.(a) Determine the mean duration of a typical sojourn in state 0.(b) Using renewal theory, determine the long run fraction of time that the
7.4.4 A developing country is attempting to control its population growth by placing restrictions on the number of children each family can have. This society places a high premium on female children, and it is felt that any policy that ignores the desire to have female children will fail. The
7.4.3 Suppose that the life of a lightbulb is a random variable X with hazard rate h.x/ D x for x > 0. Each failed lightbulb is immediately replaced with a new one. Determine an asymptotic expression for the mean age of the lightbulb in service at time t, valid for t0.
7.4.2 A system is subject to failures. Each failure requires a repair time that is exponentially distributed with rate parameter . The operating time of the system until the next failure is exponentially distributed with rate parameter . The repair times and the operating times are all
7.4.1 Suppose that a renewal function has the form M.t/ D tC[1????exp.????at/]. Determine the mean and variance of the interoccurrence distribution.
7.4.6 A machine can be in either of two states: “up” or “down.” It is up at time zero and thereafter alternates between being up and down. The lengths X1;X2; : : :of successive up times are independent and identically distributed random variables with mean , and the lengths Y1;Y2; : : : of
7.4.5 What is the limiting distribution of excess life when renewal lifetimes have the uniform density f .x/ D 1, for 0 < x < 1?
7.4.4 Show that the optimal age replacement policy is to replace upon failure alone when lifetimes are exponentially distributed with parameter . Can you provide an intuitive explanation?
7.4.3 Consider the triangular lifetime density function f .x/ D 2x, for 0 < x < 1. Determine the optimal replacement age in an age replacement model with replacement cost K D 1 and failure penalty c D 4 (cf. the example in Section 7.4.1).
7.4.2 Consider the triangular lifetime density f .x/ D 2x for 0 < x < 1. Determine an asymptotic expression for the probability distribution of excess life. Using this distribution, determine the limiting mean excess life and compare with the general result following equation (7.21).
7.4.1 Consider the triangular lifetime density f .x/ D 2x for 0 < x < 1. Determine an asymptotic expression for the expected number of renewals up to time t.Hint: Use equation (7.17).
7.3.5 Birds are perched along a wire as shown according to a Poisson process of rate per unit distance:At a fixed point t along the wire, let D.t/ be the random distance to the nearest bird. What is the mean value of D.t/? What is the probability density function ft.x/ for D.t/? 0 -D()+
7.3.4 This problem is designed to aid in the understanding of length-biased sampling.Let X be a uniformly distributed random variable on [0; 1]. Then, X divides[0; 1] into the subintervals [0;X] and .X; 1]. By symmetry, each subinterval has mean length 1 2 . Now pick one of these subintervals at
7.3.3 Pulses arrive at a counter according to a Poisson process of rate . All physically realizable counters are imperfect, incapable of detecting all signals that enter their detection chambers. After a particle or signal arrives, a counter must recuperate, or renew itself, in preparation for the
7.3.2 A fundamental identity involving the renewal function, valid for all renewal processes, isSee equation (7.7). Evaluate the left side and verify the identity when the renewal counting process is a Poisson process. E[WND+1] =E[X](M(t)+1).
7.3.1 In another form of sum quota sampling (see Chapter 5, Section 5.4.2), a sequence of nonnegative independent and identically distributed random variables X1;X2; : : : is observed, the sampling continuing until the first time that the sum of the observations exceeds the quota t. In renewal
7.3.3 Let W1;W2; : : : be the event times in a Poisson process fN.t/I t 0g of rate .Show thatare independent random variables by evaluating N(t) and WN()+1
7.3.2 Particles arrive at a counter according to a Poisson process of rate . An arriving particle is recorded with probability p and lost with probability 1????p independently of the other particles. Show that the sequence of recorded particles is a Poisson process of rate p.
7.3.1 Let W1;W2; : : : be the event times in a Poisson process fX.t/I t 0g of rate .Evaluate Pr{WN()+1>+s} and E[WN()+1].
7.2.3 Determine M.n/ when the interoccurrence times have the geometric distribution Pr{X =k} pk (1-8)-1 for k = 1, 2,..., where 0
7.2.2 Let X1;X2; : : : be the interoccurrence times in a renewal process. Suppose PrfXk D 1g D p and PrfXk D 2g D q D 1????p. Verify thatfor n D 2;4;6; : : : : n q M(n) =E[N(n)] = 1+9 (1+9) [1-(-9)"]
7.2.1 For the block replacement example of this section for which p1 D 0:1;p2 D 0:4;p3 D 0:3, and p4 D 0:2, suppose the costs are c1 D 4 and c2 D 5. Determine the minimal cost block period K and the cost of replacing upon failure alone.
7.2.3 Calculate the mean number of renewalsM.n/ D E[N.n/] for the renewal process having interoccurrence distribution p1 D 0:4; p2 D 0:1; p3 D 0:3; p4 D 0:2 for n D 1;2; : : : ; 10. Also calculate un D M.n/????M.n????1/.
7.2.2 A certain type component has two states: 0 D OFF and 1 D OPERATING. In state 0, the process remains there a random length of time, which is exponentially distributed with parameter , and then moves to state 1. The time in state 1 is exponentially distributed with parameter , after which the
7.2.1 Let fXnIn D 0;1; : : :g be a two-state Markov chain with the transition probability matrixState 0 represents an operating state of some system, while state 1 represents a repair state. We assume that the process begins in state X0 D 0, and then the successive returns to state 0 from the
7.1.4 Let t be the excess life and t the age in a renewal process having interoccurrence distribution function F.x/. Determine the conditional probability Prft > yjt D xg and the conditional mean E[tjt D x].
7.1.3 A fundamental identity involving the renewal function, valid for all renewal processes, is E[WN.t/C1] D E[X1].M.t/C1/:See equation (7.7). Using this identity, show that the mean excess life can be evaluated in terms of the renewal function via the relation E[ t] D E[X1].1CM.t//????t:
7.1.2 From equation (7.5), and for k 1, verify thatand carry out the evaluation when the interoccurrence times are exponentially distributed with parameter , so that dFk is the gamma density Pr{N(t) k) Pr{Wk 1 < Wk+1} t 1-F(t-x)]dFx(x), 0
7.1.1 Verify the following equivalences for the age and the excess life in a renewal process N.t/: (Assume t > x.)Carry out the evaluation when the interoccurrence times are exponentially distributed with parameter , so that dFk is the gamma density Pr{8 x, y > y} = Pr{N(t-x) = N(t+y)} =Pr{Wk
7.1.4 Consider a renewal process for which the lifetimes X1;X2; : : : are discrete random variables having the Poisson distribution with mean . That is,(a) What is the distribution of the waiting time Wk?(b) Determine PrfN.t/ D kg. e-n Pr{Xk=n)= for n = 0, 1,.... n!
7.1.3 Which of the following are true statements?(a) N.t/ < k if and only if Wk > t.(b) N.t/ k if and only if Wk t.(c) N.t/ > k if and only if Wk < t.
7.1.2 Consider a renewal process in which the interoccurrence times have an exponential distribution with parameter :f .x/ D e????x; and F.x/ D 1????e????x for x > 0:Calculate F2.t/ by carrying out the appropriate convolution [see the equation just prior to (7.3)], and then determine PrfN.t/ D
7.1.1 Verify the following equivalences for the age and the excess life in a renewal process N.t/:t > x if and only if N.tCx/????N.t/ D 0I and for 0 < x < t,t > x if and only if N.t/????N.t????x/ D 0:Why is the condition x < t important in the second case but not the first?

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