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probability and stochastic modeling

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

6.7.12 Let be the time of the first event in a .0;/ Cox process. Let the 0 and states represent “OFF” and “ON,” respectively.(a) Show that the total duration in the .0; ] interval that the system is ON is exponentially distributed with parameter and does not depend on ;;or the starting
6.7.11 A Stop-and-Go Traveler The velocity V.t/ of a stop-and-go traveler is described by a two-state Markov chain. The successive durations in which the traveler is stopped are independent and exponentially distributed with parameter, and they alternate with independent exponentially distributed
6.7.10 Consider a stationary .0;/ Cox process.(a) Show thatwhence(b) Establish the limitwhere(c) We interpret the limit in (b) as the conditional probability PrfN..0; t]/ D 0jEvent occurs at time 0g:Show thatwhere(d) Let be the time to the first event in .0;1/ in a stationary .0;/ Cox process with
6.7.9 Show thatandwithsatisfy the differential equations (6.77) and (6.78) subject to the initial conditions (6.79) and (6.80). fo(t)=a+e++a_e="-1
6.7.8 Consider a stationary .0;/ Cox process. A long duration during which no events were observed would suggest that the intensity process is in state 0.Show thatwhere f0.t/ is defined in (6.75). fo(t) Pr{(t)=0|N((0,1]) = 0} = f(t)
6.7.7 Consider a .0;/ stationary Cox process with D D 1 and D 2. Show that g.tI / D f .tI .1????// is given bywhereUse this to evaluate PrfN..0; 1]/ D 1g. 8(1, 0) =-(2-6 g(1; e-(2-cosh(Rt) + cosh(Rt)+=sinh(Rt) | 1
6.7.6 Show that the Laplace transformis given bywhere D C and D =. C/. Evaluate the limit (a) as !1, and (b)as !0. 2) (s; )= [ef(t; )dt 0
6.7.5 Determine the conditional probability of no points in the interval .t; tCs], given that there are no points in the interval .0; t] for a stationary Cox process driven by a two-state Markov chain. Establish the limit lim Pr{N((t,t+s])=0|N((0, ]) = 0) = e-s, s>0. 100
6.7.4 Let T be the time to the first event in a stationary .0;/ Cox process. Find the expected value E[T]. Show that E[T] D 3 2 when D D 1 and D 2. What is the average duration between events in this process?
6.7.3 Let T be the time to the first event in a stationary .0;/ Cox process. Find the probability density function .t/ for T. Show that when D D 1 and D 2, this density function simplifies to .t/ D expf????2tgcosh.p 2t/.
6.7.2 The excess life .t/ in a point process is the random length of the duration from time t until the next event. Show that the cumulative distribution function for the excess life in a Cox process is given by Prf .t/ xg D 1????PrfN..t; tC x]/ D 0g.
6.7.1 Consider a stationary Cox process driven by a two-state Markov chain. Let D =. C/ be the probability that the process begins in state .(a) By using the transition probabilities given in (6.30a–d), show that Prf.t/ D g D for all t > 0.(b) Show that E[N..0; t]/] D t for all t > 0.
6.7.2 Suppose that a .0;/ Cox process has D D 1 and D 2. Show thatsatisfy the differential equations (6.77) and (6.78) with the initial conditions (6.79) and (6.80). 1+2 fo(t) -(2-2)1+. -(2+2)1 4 4 and fit) = e(2v2)r 4 -(2+2)1 +-e 4
6.7.1 Suppose that a .0;/ Cox process has D D 1 and D 2. Show that D 2p 2 and c???? D 14.2C p2/ D 1????cC, whence -21 Pr{N((0, ])=0} e [cosh (2t) + 1/2 sin - sinh(V2)
6.6.4 A system consists of two units, both of which may operate simultaneously, and a single repair facility. The probability that an operating system will fail in a short time interval of length 1t is .1t/Co.1t/. Repair times are exponentially distributed, but the parameter depends on whether the
6.6.3 Let X1.t/;X2.t/; : : : , XN.t/ be independent two-state Markov chains having the same infinitesimal matrixDetermine the infinitesimal matrix for the Markov chain Z.t/ D X1.t/C C XN.t/. 0 1 0-2 A|= 1 -
6.6.2 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
6.6.1 Let Yn;n D 0;1; : : : ; be a discrete time Markov chain with transition probabilities P D kPijk, and let fN.t/I t 0g be an independent Poisson process of rate .Argue that the compound process X.t/ D YN.t/; t 0;is a Markov chain in continuous time and determine its infinitesimal
6.6.2 Let X1.t/ and X2.t/ be independent two-state Markov chains having the same infinitesimal matrixArgue that Z.t/ D X1.t/CX2.t/ is a Markov chain on the state space S D f0;1;2g and determine the transition probability matrix P.t/ for Z.t/. 01 0-
6.6.1 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
6.5.2 Consider a birth and death process on the states 0;1; : : : ;5 with parametersNote that 0 and 5 are absorbing states. Suppose the process begins in state X.0/ D 2.(a) What is the probability of eventual absorption in state 0?(b) What is the mean time to absorption? = = = 5 = 0, = 1, 12 =
6.5.1 Consider the sterile male control model as described in the example entitled“Sterile Male Insect Control ” and let um be the probability that the population becomes extinct before growing to size K starting with X.0/ D m individuals.Show that um=- where im Pi K-1 i-0 Pi for m = 1,..., K,
6.5.2 Assume that > 1 and verify the following steps in the approximation to Mg, the mean generation to extinction as given in (6.53): K = K -ok-i+1 i=1 OK- K K-j+1 2 K K K-1 K + + + K K-1 K-2 K 1 = 1-(1/0). K(0-1)
6.5.1 Assuming Mg= K K = 0xdx i=10 In -0 1 1-x^ =0-1 dx -dx = 0-1 dx=0 -dx 1-x 1-x 0 1-x -0-1xx 0 x*(1+x+x+...)dx 10K+1 OK+2 In + +. 1-0 0K+1 K+2 K+1 K+1 In 1+ 0+ 02 1-0 K+1 K+2 K+3 2 + ...). 1 22 -In 1-0 (K+1)(1-0)
6.4.8 A birth and death process has parametersandAssuming that ka(k+1) for k = 0, 1, 2,...,
6.4.7 A system consists of three machines and two repairmen. At most two machines can operate at any time. The amount of time that an operating machine works before breaking down is exponentially distributed with mean 5. The amount of time that it takes a single repairman to fix a machine is
6.4.6 A time-shared computer system has three terminals that are attached to a central processing unit (CPU) that can simultaneously handle at most two active users.If a person logs on and requests service when two other users are active, then the request is held in a buffer until it can receive
6.4.5 A chemical solution contains N molecules of type A and an equal number of molecules of type B. A reversible reaction occurs between type A and B molecules in which they bond to form a new compound AB. Suppose that in any small time interval of length h, any particular unbonded A molecule will
6.4.4 This problem considers a continuous time Markov chain model for the changing pattern of relationships among members in a group. The group has four members: a;b;c, andd. Each pair of the group may or may not have a certain relationship with each other. If they have the relationship, we say
6.4.3 A factory has five machines and a single repairman. The operating time until failure of a machine is an exponentially distributed random variable with parameter(rate) 0.20 per hour. The repair time of a failed machine is an exponentially distributed random variable with parameter (rate) 0.50
6.4.2 Determine the stationary distribution, when it exists, for a birth and death process having constant parameters n D for n D 0;1; : : : and n D for n D 1;2; : : : :
6.4.1 For the repairman model of the second example of this section, suppose that M D N D 5;R D 1; D 2, and D 1. Using the limiting distribution for the system, determine(a) The average number of machines operating.(b) The equipment utilization.(c) The average idle repair capacity.How do these
6.4.6 A birth and death process has parameters n D and n D n, for n D 0;1; : : : :Determine the stationary distribution.
6.4.5 Consider the birth and death parameters n D < 1 and n D n=.nC1/ for n D 0;1; : : : : Determine the stationary distribution.
6.4.4 Consider two machines, operating simultaneously and independently, where both machines have an exponentially distributed time to failure with mean 1=( is the failure rate). There is a single repair facility, and the repair times are exponentially distributed with rate .(a) In the long run,
6.4.3 Determine the stationary distribution for a birth and death process having infinitesimal parameters n D .nC1/ and n D n2 for n D 0;1; : : : ; where 0 < < .
6.4.2 Let X.t/ be a birth and death process where the possible states are 0;1; : : : ;N, and the birth and death parameters are, respectively, n D .N ????n/ andn D n. Determine the stationary distribution.
6.4.1 In a birth and death process with birth parameters n D for n D 0;1; : : : and death parameters n D n for n D 0;1; : : : ; we havewhereVerify that these transition probabilities satisfy the forward equations (6.34), with i D 0. Poj(t)= (ap)le-Ap j!
6.3.4 A Stop-and-Go Traveler The velocity V.t/ of a stop-and-go traveler is described by the two-state Markov chain whose transition probabilities are given by(6.30a–d). The distance traveled in time t is the integral of the velocity:Assuming that the velocity at time t D 0 is V.0/ D 0, determine
6.3.3 Let fV.t/g be the two-state Markov chain whose transition probabilities are given by (6.30a–d). Suppose that the initial distribution is .1????;/. That is, assume that PrfV.0/ D 0g D 1???? and PrfV.0/ D 1g D . For 0 E[V(s)V(t)]=-10(-s), whence Cov[V(s), V(t)]=(1-)(a+)\-s|
6.3.2 Collards were planted equally spaced in a single row in order to provide an experimental setup for observing the chaotic movements of the flea beetle (Phyllotreta cruciferae). A beetle at position k in the row remains on that plant for a random length of time having mean mk (which varies with
6.3.1 Let n;n D 0;1; : : : , be a two-state Markov chain with transition probability matrixLet fN.t/I t 0g be a Poisson process with parameter . Show that X.t/ D N.t/; t 0;is a two-state birth and death process and determine the parameters 0 and 1 in terms of and . P= 01 0 11-a 1
6.3.3 Let fV.t/g be the two-state Markov chain whose transition probabilities are given by (6.30a–d). Suppose that the initial distribution is .1????;/. That is, assume that PrfV.0/ D 0g D 1???? and PrfV.0/ D 1g D . In this case, show that PrfV.t/ D 1g D for all times t > 0.
6.3.2 Patients arrive at a hospital emergency room according to a Poisson process of rate . The patients are treated by a single doctor on a first come, first served basis. The doctor treats patients more quickly when the number of patients waiting is higher. An industrial engineering time study
6.3.1 Particles are emitted by a radioactive substance according to a Poisson process of rate . Each particle exists for an exponentially distributed length of time, independent of the other particles, before disappearing. Let X.t/ denote the number of particles alive at time t. Argue that X.t/ is
6.2.6 Let T be the time to extinction in the linear death process with parameters X.0/ D N and (see Section 6.2.1). (a) Using the sojourn time viewpoint, show that(b) Verify the result of (a) by using equation (6.15) in E[T] == [+ N-1 + + }]
6.2.5 Consider a cable composed of fibers following the breakdown rule K[l] D sinh.l/ D 1 2????el ????e????lfor l 0. Show that the mean cable life is given by E[Wv]= {ksinh(NL/k)) 1 = 2 k=1 0 N sinh k/N {xsinh (L/x))dx.
6.2.4 A chemical solution contains N molecules of type A and M molecules of type B. An irreversible reaction occurs between type A and B molecules in which they bond to form a new compound AB. Suppose that in any small time interval of length h, any particular unbonded A molecule will react with
6.2.3 A pure death process X.t/ with parameters 1;2; : : : starts at X.0/ D N and evolves until it reaches the absorbing state 0. Determine the mean area under the X.t/ trajectory.Hint: This is E[W1 CW2 C CWN].
6.2.2 Let X.t/ be a pure death process with constant death rates k D for k D 1;2; : : : ;N. If X.0/ D N, determine Pn.t/ D PrfX.t/ D ng for n D 0;1; : : : ;N.
6.2.1 Let X.t/ be a pure death process starting from X.0/ D N. Assume that the death parameters are 1;2; : : : ;N. Let T be an independent exponentially distributed random variable with parameter . Show that N Pr{X(T) = 0} = [] Mi+0 i=1
6.2.4 Consider the linear death process (Section 6.2.1) in which X.0/ D N D 5 and D 2. Determine PrfX.t/ D 2g.Hint: Use equation (6.14).
6.2.3 Give the transition probabilities for the pure death process described by X.0/ D 3;3 D 1;2 D 2, and 1 D 3.
6.2.2 A pure death process starting from X.0/ D 3 has death parameters 0 D 0;1 D 3;2 D 2, and 3 D 5. Let W3 be the random time that it takes the process to reach state 0.(a) Write W3 as a sum of sojourn times and thereby deduce that the mean time is E[W3] D 31 30 .(b) Determine the mean of W1
6.2.1 A pure death process starting from X.0/ D 3 has death parameters 0 D 0;1 D 3;2 D 2, and 3 D 5. Determine Pn.t/ for n D 0;1;2; 3.
6.1.13 Using (6.5), derive Pn.t/ when all birth parameters are the same constant and show thatThus, the postulates of Section 6.1.1 serve to define the Poisson processes. Pn(t)= (At)"e-At n! n = 0,1,....
6.1.12 Verify that P2.t/, as given by (6.8), satisfies (6.5) by following the calculations in the text that showed that P1.t/ satisfies (6.5).
6.1.11 Beginning with P0.t/ D e????0t and using equation (6.5), calculate P1.t/;P2.t/, and P3.t/ and verify that these probabilities conform with equation (6.7), assuming distinct birth parameters.
6.1.10 Consider a pure birth process on the states 0;1; : : : ; N for which k D .N ????k/for k D 0;1; : : : ;N. Suppose that X.0/ D 0. Determine Pn.t/ D PrfX.t/ D ng for n D 0; 1, and 2.
6.1.9 Under the conditions of Problem 6.8, determine E[N.t/].
6.1.8 Let N.t/ be a pure birth process for which Prfan event happens in.t; tCh/jN.t/ is oddg D hCo.h/;Prfan event happens in.t; tCh/jN.t/ is eveng D hCo.h/;where o.h/=h!0 as h # 0. Take N.0/ D 0. Find the following probabilities:P0.t/ D PrfN.t/ is evengI P1.t/ D PrfN.t/ is oddg:Hint: Derive the
6.1.7 Let 0;1, and 2 be the parameters of the independent exponentially distributed random variables S0;S1, and S2. Assume that no two of the parameters are equal.(a) Verify thatand evaluate in similar terms(b) Verify equation (6.8) in the case that n = 2 by evaluating Pr{So > 1}=e-hot Pr{So+S>t} =
6.1.6 A fatigue model for the growth of a crack in a discrete lattice proposes that the size of the crack evolves as a pure birth process with parametersThe theory behind the model postulates that the growth rate of the crack is proportional to some power of the stress concentration at its ends and
6.1.5 Let Wk be the time to the kth birth in a pure birth process starting from X.0/ D 0. Establish the equivalenceFrom this relation together with equation (6.7), determine the joint density for W1 and W2, and then the joint density of S0 D W1 and S1 D W2 ????W1. Pr{Wt, Wt+s} = Po(t)[Po(s) +
6.1.4 A new product (a “Home Helicopter” to solve the commuting problem)is being introduced. The sales are expected to be determined by both media (newspaper and television) advertising and word-of-mouth advertising, wherein satisfied customers tell others about the product. Assume that media
6.1.3 Consider a population comprising a fixed number N of individuals. Suppose that at time t D 0, there is exactly one infected individual and N ????1 susceptible individuals in the population. Once infected, an individual remains in that state forever. In any short time interval of length h, any
6.1.2 A Yule process with immigration has birth parameters k D Ck for k D 0;1;2; : : : : Here, represents the rate of immigration into the population, and represents the individual birth rate. Supposing that X.0/ D 0, determine Pn.t/for n D 0;1;2; : : : :
6.1.1 Let X.t/ be a Yule process that is observed at a random time U, where U is uniformly distributed over [0; 1). Show that PrfX.U/ D kg D pk=.k/ for k D 1;2; : : : , with p D 1????e????.Hint: Integrate (6.10) over t between 0 and 1.
6.1.6 Operations 1, 2, and 3 are to be performed in succession on a major piece of equipment. Operation k takes a random duration Sk that is exponentially distributed with parameter k for k D 1;2; 3, and all operation times are independent.Let X.t/ denote the operation being performed at time t,
6.1.5 Using equation (6.10), calculate the mean and variance for the Yule process where X.0/ D 1.
6.1.4 Consider an experiment in which a certain event will occur with probability h and will not occur with probability 1????h, where is a fixed positive parameter and h is a small .h / positive variable. Suppose that n independent trials of the experiment are carried out, and the total number of
6.1.3 A population of organisms evolves as follows. Each organism exists, independent of the other organisms, for an exponentially distributed length of time with parameter , and then splits into two new organisms, each of which exists, independent of the other organisms, for an exponentially
6.1.2 A pure birth process starting from X.0/ D 0 has birth parameters 0 D 1;1 D 3;2 D 2, and 3 D 5. Let W3 be the random time that it takes the process to reach state 3.(a) Write W3 as a sum of sojourn times and thereby deduce that the mean time is E[W3] D 11 6 .(b) Determine the mean of W1
6.1.1 A pure birth process starting from X.0/ D 0 has birth parameters 0 D 1;1 D 3;2 D 2, and 3 D 5. Determine Pn.t/ for n D 0;1;2; 3.
5.6.10 A Bidding Model Let U1;U2; : : : be independent random variables, each uniformly distributed over the interval .0; 1]. These random variables represent successive bids on an asset that you are trying to sell, and that you must sell by time t D 1, when the asset becomes worthless. As a
5.6.9 Let W1;W2; : : : be the event times in a Poisson process of rate , and let N.t/ D N..0; t]/ be the number of points in the interval .0; t]. EvaluateNote: 60 kD1.Wk/2 D 0. N(D) E(Wk) IM k=1
5.6.8 Let fN.t/I t 0g be a nonhomogeneous Poisson process of intensity .t/;t > 0, and let Y1;Y2; : : : be independent and identically distributed nonnegative random variables with cumulative distribution functionLetDetermine = G(y) y for 0 < y < 1.
5.6.7 Let fN.t/I t 0g be a Poisson process of intensity , and let Y1;Y2; : : : be independent and identically distributed nonnegative random variables with cumulative distribution functionDetermine PrfZ.t/ > zjN.t/ > 0g, where Z.t/ D min Y1;Y2; : : : ;YN.t/:Describe the behavior for large t.
5.6.6 LetW1;W2; : : : be the event times in a Poisson process fX.t/I t 0g of rate . A new point process is created as follows: Each point Wk is replaced by two new points located at Wk CXk and Wk CYk, where X1;Y1; X2;Y2; : : : are independent and identically distributed nonnegative random
5.6.5. Let fX.t/I t 0g and fY.t/I t 0g be independent Poisson processes with respective parameters and . Let T D minft 0IY.t/ D 1g be the random time of the first event in the Y process. Determine PrfX.T=2/ D kg for k D 0;1; : : : :
5.6.4 Let fX.t/I t 0g and fY.t/I t 0g be independent Poisson processes with respective parameters and . For a fixed integera, let Ta D minft 0IY.t/ D ag be the random time that the Y process first reaches the valuea. Determine PrfX.Ta/ D kg for k D 0;1; : : : :Hint: First consider D
5.6.3 Shocks occur to a system according to a Poisson process of intensity . Each shock causes some damage to the system, and these damages accumulate. Let N.t/ be the number of shocks up to time t, and let Yi be the damage caused by the ith shock. Thenis the total damage up to time t. Suppose that
5.6.2 Suppose that particles are distributed on the surface of a circular region according to a spatial Poisson process of intensity particles per unit area. The polar coordinates of each point are determined, and each angular coordinate is shifted by a random amount, with the amounts shifted for
5.6.1 Suppose that points are distributed over the half line [0;1) according to a Poisson process of rate . A sequence of independent and identically distributed nonnegative random variables Y1;Y2; : : : is used to reposition the points so that a point formerly at location Wk is moved to the
5.6.5 Alpha particles are emitted from a fixed mass of material according to a Poisson process of rate . Each particle exists for a random duration and is then annihilated. Suppose that the successive lifetimes Y1;Y2; : : : of distinct particles are independent random variables having the common
5.6.4 Men and women enter a supermarket according to independent Poisson processes having respective rates of two and four per minute.(a) Starting at an arbitrary time, what is the probability that at least two men arrive before the first woman arrives?(b) What is the probability that at least two
5.6.3 Let fN.t/I t 0g be a Poisson process of intensity , and let Y1;Y2; : : : be independent and identically distributed nonnegative random variables with cumulative distribution function G.y/ D PrfY yg. Determine PrfZ.t/ > zjN.t/ > 0g, where Z.t/ D minY1;Y2; : : : ;YN.t/:
5.6.2 Shocks occur to a system according to a Poisson process of intensity . Each shock causes some damage to the system, and these damages accumulate. Let N.t/ be the number of shocks up to time t, and let Yi be the damage caused by the ith shock. Then X.t/ D Y1 C CYN.t/is the total damage up
5.6.1 Customers demanding service at a central processing facility arrive according to a Poisson process of intensity D 8 per unit time. Independently, each customer is classified as high priority with probability D 0:2, or low priority with probability 1???? D 0:8. What is the probability that
5.5.7 Consider a collection of circles in the plane whose centers are distributed according to a spatial Poisson process with parameter jAj, where jAj denotes the area of the set A. (In particular, the number of centers .A/ in the set A follows the distribution law Prf.A/ D kg D e????jA .jAj/k=k!.)
5.5.6 Suppose that stars are distributed in space following a Poisson point process of intensity . Fix a star alpha and let R be the distance from alpha to its nearest neighbor. Show that R has the probability density function f(x) = (42.7.x) exp{4x]. -4. x>0. 3
5.5.5 Consider a two-dimensional Poisson process of particles in the plane with intensity parameter . Determine the distribution FD.x/ of the distance between a particle and its nearest neighbor. Compute the mean distance.
5.5.4 Consider spheres in three-dimensional space with centers distributed according to a Poisson distribution with parameter jAj, where jAj now represents the volume of the set A. If the radii of all spheres are distributed according to F.r/with density f .r/ and finite third moment, show that
5.5.3 LetN.A/I A 2 R2 be a homogeneous Poisson point process in the plane, where the intensity is . Divide the .0; t].0; t] square into n2 boxes of side length d D t=n. Suppose there is a reaction between two or more points whenever they are located within the same box. Determine the
5.5.2 Points are placed on the surface of a circular disk of radius one according to the following scheme. First, a Poisson distributed random variable N is observed. If N D n, then n random variables 1; : : : ; n are independently generated, each uniformly distributed over the interval [0;2/,
5.5.1 A piece of a fibrous composite material is sliced across its circular cross section of radius R, revealing fiber ends distributed across the circular area according to a Poisson process of rate 100 fibers per cross section. The locations of the fibers are measured, and the radial distance of
5.5.3 Defects (air bubbles, contaminants, chips) occur over the surface of a varnished tabletop according to a Poisson process at a mean rate of one defect per top. If two inspectors each check separate halves of a given table, what is the probability that both inspectors find defects?
5.5.2 Customer arrivals at a certain service facility follow a Poisson process of unknown rate. Suppose it is known that 12 customers have arrived during the first 3 h. Let Ni be the number of customers who arrive during the ith hour, i D 1;2; 3. Determine the probability that N1 D 3;N2 D 4, and N3
5.5.1 Bacteria are distributed throughout a volume of liquid according to a Poisson process of intensity D 0:6 organisms per mm3. A measuring device counts the number of bacteria in a 10 mm3 volume of the liquid. What is the probability that more than two bacteria are in this measured volume?
5.4.11 Computer Challenge Let U0;U1; : : : be independent random variables, each uniformly distributed on the interval .0;1/. Define a stochastic process fSng recursively by setting(This is an example of a discrete-time, continuous-state, Markov process.)When n becomes large, the distribution of Sn

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