The following information is repeated here for your convenience.Let (X;t >0} be a continuous time, time homogeneous, Markov process with state-space (1, 2, 3, 4, 5} and the following generator matrix Q. Missingentries in Q are denoted '??.-0.30.0.20.10.-0.6????0.5-0.5-0.10.1-0.4From state 2, the process is twice as likely to go to state 4 next as it is to go to state 3 next.0.4If Xo-2 compute the expected time until the process enters a closed class,Answer:

denotes a continuous time, time homogenous.The state space of the Markov process: The generator…

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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5. Find the limit state of the Markov process model by the matrix A = A=1 of Find the eig. vector si [0.7 0.21 10.3 0.81
1. Consider a Markov chain X matrix = (Xn)n≥0 with state space S [0.2 0.4 0 0.4 0.3 0 0.7 0 0.5 0 0.5 0 0 0.1 0.9 0 (a) Which states are transient and which are recurrent? (b) Is this markov chain irreducible or reducible? = {1, 2, 3, 4} and transition
The given matrix is an absorbing stochastic matrix in standard form. Identify R and S and compute the fundamental matrix and the stable matrix 1 0 0 0.1 0.2 0 1 0 0.1 0.2 0 0 1 0 0.1 0 0 0 0.5 0 0 0 0.3 0.5 0.5 R= (Simplify your answer.) 0.3 0.5 0.1 0.2 S= 0.1 0.2 (Simplify your answer.) 0 0.1 The fundamental matrix is (Simplify your answer.)
Consider the migration (Markov) matrix A = 0.7 0 0.2 0.1 0.4 0.2 0.2 0.6 0.6 Suppose that, initially, there are 96 residents in location 1, 82 residents in location 2, and 151 residents in location 3. Assume time is measured in years. Find the population in each location after 1 year. Find the population in each location after 2 years.
• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = y} is the first time after 0 that the chain visits state y. Px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
Suppose qo = [1/4, 3/4] is the initial state distribution for a Markov process with the following transition matrix: %3D [1/2 1/2] M = [3/4 1/4] (a) Find q1, 92, and q3. (b) Find the vector v that qo M" approaches. (c) Find the matrix that M" approaches.
Determine whether or not the following sets S of 2 x 2 matrices are linearly independent. 25 -4 {(²5) (22)} 0 Select an Answer Select an Answer Select an Answer Select an Answer 1. S ✓2. S 3. S 4. S = = " = 25 {(²5). ( 22 )} -4 -10 -12 0 25 -4 -22 17 -31 -{(² {(5)). (1 3). (1 3) (5 2). (772)) ¹)} " -2 0 9 10 -10 -2 ㅠ e² = {(²9)· (42) (43)} 2 3 >
Suppose in a scatterplot matrix, you observe that all of the scatterplots associated with the explanatory variables show a strong linear relationship. There are three explanatory variables in the model. Of the following, which is the most valid to do? Remove the response variable. Remove only one of the explanatory variables and keep only two. Remove all of the explanatory variables because they are linearly related to each other and therefore explain the same thing. Remove exactly two of the explanatory variables because they are all linearly related to each other and therefore explain the same thing. We only need to keep one in the model.
A beauty salon employs three hairdressers. All customers get their hair washed in one area of the saloon before being taken to another area to get their hair cut. Customer arrival rate is considered deterministic (appointment only) and service time follows the exponential distribution. What is the Kendall notation for this system? O D/M/3 OM/G/3 O M/M/1 OM/M/3
E5.2. Consider a Markov chain with state space S = {1,2,3, 4, 5, 6} and transition matrix 1 0 0 0 0 0 0 0 0 P = 0 0 0 1 0 0 0 0 0 0 0 1 0 (a) Draw the transition graph of this chain. (b) Show that state 1 is absorbing and conclude that it is positive re- current. (c) Show that states 2 and 3 are transient. (d) Verify that states 4, 5 and 6 are positive recurrent.
Classify the Markov chain in Figure 1 and calculate the periodicity of every state. 0 10 1 30 1 334 1 02 0 4 2 1 05 Figure 1: Diagram of the Markov chain of Problem 3.
Q6 You have two machines. Machine 1 has a lifetime T which is exponentially distributed with parameter d1, and Machine 2 has a lifetime T, which is exponentially distributed with parameter A, and is independent of T1. Machine 1 starts at time 0, and Machine 2 starts at a time T > 0. Q6(i.) Assume that T is deterministic. Compute the probability that Machine 1 is the first to fail. Q6(ii.) Answer the same question when T is exponentially distributed with parameter u and is independent of T1 and T,.

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