Stochastic Processes 2nd Edition Sheldon M. Ross - Solutions

Unlock comprehensive insights into "Stochastic Processes 2nd Edition" by Sheldon M. Ross with our expertly crafted solutions. Dive into a rich repository of answers key, featuring step-by-step answers and solved problems. Our solution manual is a valuable resource for students and instructors alike, providing detailed chapter solutions and a robust test bank. Whether you're seeking textbook solutions or an instructor manual, our platform offers a seamless experience. Access solutions in PDF format for free download, and explore questions and answers that enhance your understanding of this subject. Perfect for online learners seeking clarity and success.

Define Stochastic Modeling
Suppose that W is the sum of independent Bernoulli random variables. Bound the difference between P{W =k} and its approximation as given in Equation (10.3.1). Compare this bound with the one obtained in Example 10.2(A) for the usual Poisson approximation
Also determine the bound provided by Corollary 10.3.2.
10.8. Use the approximation of Section 10 3 to approximate P(system failure) in Problem
Given a set of n vertices suppose that between each of the (2) pairs of vertices there is (independently) an edge with probability p. Call the sets of vertices and edges a graph. Say that any set of three vertices constitutes a triangle if the graph contains all of the (2) edges connecting these
(c) Determine the bound on P{W = 0} that is provided by Corollary 10.3 2. (d) For N = 20, n = 3, compute the exact probability and compare with the preceding bound and approximations.
Suppose that N people toss their hats in a circle and then randomly make selections Let W denote the number of the first n people that select their own hat (a) Approximate P{W = 0} by the usual Poisson approximation. (b) Approximate P{W = 0} by using the approximation of Section
A group of 2n individuals, consisting of n couples, are randomly arranged at a round table. Let W denote the number of couples seated next to each other. (a) Approximate P{W = k} (b) Bound this approximation
If X is a Poisson random variable with mean A, show that E[Xh(X)]=E[h(X + 1)] provided these expectations exist.
Suppose that the components in Problem 10.1 are arranged in a circle. Approximate P(system failure) and use Brun's sieve to justify this ap- proximation.
Suppose that m balls are randomly chosen from a set of N, of which n are red. Let W denote the number of red balls selected. If n and m are both small in comparison to N, use Brun's sieve to argue that the distribution of W is approximately Poisson.
A set of n components, each of which independently fails with probability p, are in a linear arrangement. The system is said to fail if k consecutive components fail (a) For in+1k, let Y, be the event that components i, i + 1 all fail. Would you expect the distribution of ik - 1, Y, to be
Show that an infinite sequence of exchangeable Bernoulli random vari- ables is associated
In the model of Example 9.7(A), one can always find a family of compo- nent subsets M,,..., Mm such that the system will work if, and only if, all of the components of at least one of these subsets work (If none of the M, is a proper subset of another then they are called minimal path sets.) Show
Show that if X,.., X, are all decreasing functions of a specified set of associated random variables then they are associated.
Show that E[YX], Y. Use this to show that: (a) if X and Y are independent then XE[Y](d) X+E[X], 2X.
Suppose that P{0 X 1} = 1, and let =E[X]. Show that X , Y where Y is a Bernoulli random variable with parameter 0. (That is, P{Y 1} 0 1 - P{Y = 0}.)
If E[X] 0, show that cX 2, X when c 1
Jensen's inequality states that for a convex functionf, E[f(x)] = f(E[X]). (a) Prove Jensen's inequality(b) If X has mean E[X], show that X E[X], where E[X] is the constant random variable. (c) Suppose that there exists a random variable Z such that E[ZIY] = O and such that X has the same
Suppose P{X = 0} = 1 M/2, P{X = 2} = M/2 where 0 < M < 2. If Y is a nonnegative integer-valued random variable such that P{Y 10 and E[Y] M, then show that X , Y.
Let X,..., X be independent with P{X; = 1} = P = 1 - P{X, = 0}, i = 1, n. Show that X, Bin(n,p), 1=1 where Bin(n, ) is a binomial random variable with parameters n and Pln. (Hint Prove it first for n = 2.) Use the above to show X, Y, where Y is Poisson with mean np. Hence, for instance, a Poisson
Prove or give a counterexample to the following: If F, G, then N(t),NG(t), where {N(t) t 0} is the renewal process with interarrival distribution H, H = F, G.
Show that F is the distribution of an NBUE random variable if, and only if, F.(a) F(a) for alla, where F., the equilibrium distribution of F, is defined by Fe(a) = f* F(x) dx with = S F(x) dx
Suppose X, ..., X, are independent and Y, If X, Z, Y,, i=1,..., n, prove that E[max(X,..., X)] = E[max(Y,, , Y, are independent. Y.)]. Give a counterexample when independence is not assumed.
Show that the random variables, having the following densities, have increasing likelihood ratio; that is, the log of the densities are concave. (a) Gamma f(x) = Ae(x)/(a), a 1 (b) Weibull f(x) = a(x)-le-(x), 1 (c) Normal truncated to be positive, f(x) = 1 2 -e-(x-4)-1202 x>0.
A stockpile consists of n items Associated with the ith item is a random variable X,, i = 1, , n. If the ith item is put into the field at time 1, its field life is X,e. If X, LR X+1, i = 1,.,n 1, what ordering stochastically maximizes the total field life of all items? Note that if n = 2 and the
We have n jobs with the ith requiring a random time X, to process. The jobs must be processed sequentially and the objective is to stochastically maximize the number of jobs that are processed by a (fixed) time (a) Determine the optimal strategy if X,SX+19 LR i=1,.,n-1.(b) What if we only assumed
Consider the statistical inference problem where a random variable X is known to have density either for g The Bayesian approach is to postulate a prior probability p that f is the true density The hypothesis that were the true density would then be accepted if the posterior probability given the
A family of random variables {X, E [a, b]} is said to be a monotone likelihood ratio family if X, LR X, when 0, 02. Show that the follow- ing families have monotone likelihood ratio (a) X, is binomial with parameters n, 0, n fixed. (b) X, is Poisson with mean (c) X, is uniform (0, 0) (d) X, is
Let F and G have hazard rate functions AF and AG Show that AF(t) = (t) for all t if, and only if, there exist independent continuous random variables Y and Z such that Y has distribution G and min(Y, Z) has distribution F
Let X and Y have respective hazard rate functions Ax(t) Ay(t) for all t Define a random variable X as follows: If Y=t, then x={'_ with probability Ax(t)/Ay(t) + X, with probability 1 - Ax(t)/Ay(t), where X, is a random variable, independent of all else, with distribution P{X,>s}= P{X>+s} P{X>1}
Let X, and X2 have respective hazard rate functions A,(t) and A2(t). Show that A(1) A(t) for all t if, and only if, P{X} P{X>1} P{Xs} P{X> s} for all s
Let R, ., Rm be a random permutation of the numbers 1,.,m in the sense that for any permutation i,, im, P{R, = i,, for j = 1,., m} = 1/m' (a) In the coupon collectors problem described in Example 92(E), argue that E[N]=E[1/PRIR, is the last of types R,. ., R, to be collected], where N is the number
Verify the inequality (924) in Example 92 (E).
For random variables X and Y and any set of values A, prove the coupling bound | P{X E A} - P{YE A}| P{XY}
Consider a conditional Poisson process (see Section 2.6 of Chapter 2). That is, let A be a nonnegative random variable having distribution G and let {N(t), 0} be a counting process that, given that A = A, is a Poisson process with rate A. Let G denote the conditional distribution of A given that
Let {N,(t), t 0}, i = 1, 2, denote two renewal processes with respective interarrival distributions F, and F. Let A,(t) denote the hazard rate function of F, i = 1, 2. If A(1) A(s) for all s, t, show that for any sets T,..., T., (N(T), N(T)) = (N(T),.., N(T.)).
Consider a Markov chain with transition probability matrix P., and sup- pose that P, increases in i for all k. (a) Show that, for all increasing functionsf, , P.,f(j) increases in i. (b) Show that P, increases in i for all k, where P", are the n-step transition probabilities, n 2.
Prove that a normal distribution with mean and variance o increases stochastically as increases. What about as 2 increases?
Consider a Markov chain with states 0, 1, n and with transition probabilities P -{156 = ifj-i-1 if j = i + k(i), = (i=1,.,n 1; Poo P1),where k(i) 0. Letf, denote the probability that this Markov chain ever enters state 0 given that it starts in state i Show thatf, is an increasing function of (c),
Show that a binomial n, p distribution B, increases stochastically both as n increases and as p increases. That is, B, increases in n and in p. ....
A random variable taking on nonnegative integer values is said to be discrete IFR if P{X = k|X k} is nondecreasing in k, k = 0, 1, ... Show that (a) binomial random variables, (b) Poisson random variables, and (c) negative binomial random variables are all discrete IFR random variables. (Hint The
The following theorem concerning IFR distributions can be proven. Theorem: If F and G are IFR, then so if F * G, the convolution of F and G Show that the above theorem is not true when DFR replaces IFR
If X, i = 1,..., n, are independent IFR random variables, show that min(X,,.. X) is also IFR. Give a counterexample to show that max(X,, X) need not be. "
A random variable is said to have a gamma distribution if its density is given, for some > 0, a > 0, by where f(t) = - Ae(At)-1 (a) 10, ' T() = f e-x-1dx. == Show that this is IFR when a 1 and DFR when a 1
One of n elements will be requested-it will be i with probability Pi, i = 1,.,n If the elements are to be arranged in an ordered list,find the arrangement that stochastically minimizes the position of the element requested.
(a) If X, Y, show that P{XY}. Assume independence (b) If P {XY} , P{Y Z}, and X, Y, Z are independent, does this imply that P{XZ} Prove or give a counterexample
Suppose X, Y, i = 1, 2 Show by counterexample that it is not necessarily true that X + X2 Y + Y.
If X, Y, prove that X 2 Y and Y-Z, X-
For a second-order stationary process with mean for which = R(i)/ n0, show that for any > () 1=0 - as n
Prove the Cauchy-Schwarz inequality: n (E[XY]) E[X]E[Y]. (Hint: Start with the inequality 2|xy| x + y and then substitute XIVE[X] for x and Y/VE[Y] for y)
Show that R(i) n 2-0 implies (ji)0 KIN thus completing the proof of Proposition 8.8 1
Let U be uniformly distributed over (-, ), and let X = cos(nU). By using the trigonometric identity cos x cos y = [cos(x + y) + cos(x y)], verify that {X, n 1} is a second-order stationary process.
Verify that {X(t) = N(t + L) N(t), t = 0} is stationary when {N(t)} is a Poisson process
Show that p(x, t, y) = V2t satisfies the backward and forward diffusion Equations (8.5.1) and (852).
In Problem 816, find Var(T) by using a martingale argument.
Verify that if {B(t), t 0} is standard Brownian motion then {Y(r), t 0} is a martingale with mean 1, when Y(t) = exp{cB(t) - ct/2}
Prove that, with probability 1, for Brownian motion with drift (1) as 1
Consider Brownian motion with reflecting barriers of -B and A, A > 0, B > 0. Let p,(x) denote the density function of X,. (a) Compute a differential equation satisfied by p,(x) (b) Obtain p(x) = lim,... p(x)
Let {X(t), 0} be a Brownian motion with drift coefficient , p < 0, which is not allowed to become negative. Find the limiting distribution of X(t).
In Example 8.4(B), suppose Xox and player I has the doubling option Compute the expected winnings of I for this situation.
For a Brownian motion process with drift coefficient let f(x) =E[time to hit either A or -B|X = x], where A > 0, B>0, -B 0. (c) What is the relationship between g(x), g(y), and g(x + y) in (b) (d) Solve for g(x). (e) Verify your solution by differentiating (84.11)
Let T, denote the time until Brownian motion 'hits x Compute P{T < T < T}.
For Brownian motion with drift coefficient , show that for x>0 | X(s)|>x} = o(h). P max X(s): =
Let 7, denote the largest zero of X(r) that is less than 1 and let T be the smallest zero greater than Show that (a) P{T, 1. (b) P{T y} = (2/) arc sine sly, s
Compute the density function of T,, the time until Brownian motion hits x
Let M(1) maxsss, X(s) and show that P{M(t) > a|M(t) = X(1)} = e121 a>0
Suppose X(1) = B. Characterize, in the manner of Proposition 81.1, {X(t), 0 1) given that X(1) = B ==
Find the distribution of (a) |X(t)| (b) min X(s) 05151 (c) max X(s)X(t). 05551
Let {X(t), 0} denote a birth and death process that is allowed to go negative and that has constant birth and death rates = , = , n = 0, 1, 2, Define and c as functions of A in such a way that {cX(t), tu} converges to Brownian motion as A In Problems 87 through 8 12, let {X(t), t 0} denote Brownian
A stochastic process {X(t), 0} is said to be stationary if X(t), ... X() has the same joint distribution as X(t, + a),., X(1, +a) for all n,a, t, t (a) Prove that a necessary and sufficient condition for a Gaussian pro- cess to be stationary is that Cov(X(s), X(t)) depends only on t-s st, and
Compute the conditional distribution of X(s) given that X() X(t) =B, where t < s
Let W(t) = X(at)la for a > 0. Verify that W(r) is also Brownian motion ===
Let Y(t) 1X(1/1). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y(1)) (c) Argue that {Y(t), t 0} is also Brownian motion (d) Let T = inf{t > 0: X(t) = 0} Using (c) present an argument that P{T = 0} = 1.
Let S,, n 0, denote a random walk in which X, has distribution F Let G(t, s) denote the probability that the first value of S, that exceeds t is less than or equal to + s That is, Show that G(t, s) = P{first sum exceeding is 1 + s} - G(t, s) = F(t + s) F(t) + f G(t y, s) dF(y). 00
Let S =X, be a random walk and let A,, i > 0, denote the probability that a ladder height equals ithat is, = P{first positive value of S, equals i}. (a) Show that if 9. j= -1 P{X, = j}= 9 + = 1, j 1, then , satisfies = + 9(+1 +), i > 0.= === (b) If P{X, j, j = -2, -1, 0, 1, 2, show that = 1+5 3+ 5
For a random walk with = E[X] > 0 argue that, with probability 1, u(t) 1 it ast , where u(t) equals the number of n for which 0 S, t.
In the ruin problem of Section 7.4 let F denote the interarrival distribu- tion of claims and let G be the distribution of the size of a claim. Show that p(A), the probability that a company starting with A units of assets is ever ruined, satisfies P(A) = * *p(A + ct - x) dG(x) dF(t) + (A + ct)
In the insurance ruin problem of Section 74 explain why the company will eventually be ruined with probability 1 if E[Y] = CE[X].
Use Jensen's inequality, which states that E[f(x)] = f(E[X]) whenever f is convex to prove that if 00, E[X] < 0, and E[e] = 1, then > 0
Let S, n0 denote a random walk for which =E[SS] = 0.Let, for A > 0, B > 0, N = min{n S, A or S, -B}. Show that E[N] AB} > 0. Then show that E[N] KE[G], where G is an appropriately defined geometric random variable.)
Argue that the random walk for which X, only assumes the values 0, 1,, M and E[X] = 0 is null recurrent
An ordinary deck of cards is randomly shuffled and then the cards are exposed one at a time At some time before all the cards have been exposed you must say "next," and if the next card exposed is a spade then you win and if not then you lose For any strategy, show that at the moment you call
If X1, X2,.. is an infinite sequence of exchangeable random var- ables, with E[X] 0 (Hint: Look at Var(2X)) Give a counterexample when the set of exchangeable ran- dom variables is finite
Let X1, X2, X, be exchangeable. Compute E[XX(1), X(2), ..., X(n)], where X(1) X(2)
For the simple random walk compute the expected number of visits to state k. ..
Let X,.., X, be equally likely to be any of the n! permutations of (1, 2, ., n) Argue that P{X, a} = P{{X, = n(n + 1)/2-a}
Consider the following model for the flow of water in and out of a dam. Suppose that, during day n, Y, units of water flow into the dam from outside sources such as rainfall and river flow At the end of each day water is released from the dam according to the following rule: If the water content of
Let {X(t), t 0} be a compound Poisson process with Poisson rate and component distribution F. Define a continuous-time martingale related to this process
Let {X(1), 10} be a continuous-time Markov chain that will, in finite expected time, enter an absorbing state N Suppose that X(0) = 0 and let m, denote the expected time the chain is in state i. Show that for j + 0, j + N. (a) E[number of times the chain leaves state j] = v,m,, where 1/v, is the
Let {N(t), 1 0} be a nonhomogeneous Poisson process with intensity function (t), t 0. Let 7 denote the time at which the nth event occurs Show that n = E = E [S, X di].
Let X1, X2, be independent and such that ; P{X, -1} 1 1/2', P{X, 2 1}=1/2', i1. Use this sequence to construct a zero mean martingale Z, such that lim, Z = - with probability 1 (Hint Make use of the Borel- Cantelli lemma.) A continuous-time process {X(t), t 0} is said to be a martingale if E[|X(t)]
Consider a sequence of independent tosses of a coin and let P{head} be the probability of a head on any toss. Let A be the hypothesis that P{head} = a and let B be the hypothesis that P{head} =b, 0 1 What can we say about Z, for n large?
An urn initially contains one white and one black ball At each stage a ball is drawn and is then replaced in the urn along with another ball of the same color. Let Z, denote the fraction of balls in the urn that are white after the nth replication. (a) Show that {Zn, n 1} is a martingale. (b) Show
Let {X, n 0} be a Markov process for which X is uniform on (0, 1) and, conditional on X, (ax +1- with probability X +1 = Laxn with probability 1 - X where 0 < a
A group of 2n people, consisting of n men and n women, are to be independently distributed among m rooms Each woman chooses room j with probability p, while each man chooses it with probability q,, j 1,.., m Let X denote the number of rooms that will contain exactly one man and one woman.. = (a)
Let X, ., X, be independent random vectors that are all uniformly distributed in the circle of radius 1 centered at the origin Let T =T(X,, X,) denote the length of the shortest path connecting these n points. Argue that P{T E[T] a} 2 exp{-a/(32n)}.
For binary n-vectors x and y (meaning that each coordinate of these vectors is either 0 or 1) define the distance between them by p(x, y) = |x, y| =1 - (This is called the Hamming distance) Let A be a finite set of such vectors, and let X,. ., X, be independent random variables that are each
Let p denote the probability that a random selection of 88 people will contain at least three with the same birthday Use Azuma's inequality to obtain an upper bound on p (It can be shown that p = 50)
Suppose that 100 balls are to be randomly distributed among 20 urns Let X denote the number of urns that contain at least five balls Derive an upper bound for P{X 15}.

Showing 5400 - 5500 of 6914

First
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
Last

Stochastic Processes 2nd Edition Sheldon M. Ross - Solutions

Step by Step Answers