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 ??.-0.30.20.1-0,6????0.5-0.501-0.10.10.4-0.4From state 2, the process is twice as likely to go to state 4 next as it is to go to state 3 next.If Xo-2 compute the expected time until the process enters a closed class. Give your answer to 1 decimal place.Answer:

Let {Xt :t≥0} be a continuous time homogenous Markov process with state space {1,2,3,4,5} and the…

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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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3. A rat is put into the following maze: 3 moves. 1 2 The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n (a) Explain why {X₂} is a MC and find the transition matrix P. (b) Explain why the chain is irreducible, aperiodic and positive recurrent. (c) What is the limit of Pn? Explain. (d) Find the probability that the rat is in compartment 3 after two moves. (e) In the long run, how many times that the rat enters in compartment 4 in 100 movements?
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Suppose that a Markov chain has the following transition matrix a, az az as a s 3 000 The recurrent states are a 4 4 a₂000 P=a₂00 0 0 a003 as 1 0 0 0
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.
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• 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 >
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7
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.
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