5. Consider a finite state discrete time Markov chain {Xk} on three states S1, S2, S3 with the transitionmatrix1P =2(a) Find a stationary distribution T for P. Is it unique?(b) Is this Markov chain aperiodic? Is it irreducible?1 1(c) Suppose that xofunction at time 0. What is the probability that by time T = 4 we visit state S3?(P(Xo = S1), P(Xo = S2), P(Xo = S3))T = (},,)T is the probability mass(d) What is the probability that we visit state S2 or S3 by time T = 4?H/4112O

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A First Course in Probability (10th Edition)

10th Edition

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Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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A Markov chain {s} with state space n={1, 2, 3} has a sequence of realizations and process transitions as follows: Transition (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3.1) (3,2) (3,3) t S: S+1 2 1 1. 1 1 1 1 1 3 1 3 3 3 4 3 1. 2 3 1. 2 1 7 2 2 1. 8 2 2 1 9 2 3 1. 10 3 11 1 1 12 1 1. 13 1 1. 14 1. 1. 15 3 1. 1. 16 1. Number of frequencies : 1 3. 2 2 1 1 1 1 a. determine the transition frequency matrix O= 2 12 93 021 022 °23 %3D 31 °32 °33) 1 P1 P12 P13 b. determine the estimated transition probability matrixP= 2 P21 P22 P23 3 P31 P32 P33)
[stochastic process1]
• 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…
Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1
a)
Find the limiting distribution for this Markov chain. Then give an interpretation of what the first entry of the distribution you found tells you based on the definition of a limiting distribution. Your answer should be written for a non-mathematician and should consist of between 1 and 3 complete sentences without mathematical symbols or terminology.
Let (X0, X1, X2, . . .) be the discrete-time, homogeneous Markov chain on state space S = {1, 2, 3, 4, 5, 6} with X0 = 1 and transition matrix
c) Consider the transition Markov chain with three states S = (1, 2, 3) as shown below. %3D 1 Given that; P(X, = 1) = 3' fine P(X, = 1,X, = 2,X2 = 3) N/3 1/2 1/2 1/4 1/2 1/4
8. List the Gauss–Markov conditions required for applying a t & F-tests.
C)
Q3) The state transition matrix of a Markov random process is given by 1/3 1/3 1/6 1/6 5/9 0 0 4/9 2/5 1/5 1/5 1/5 0 0 3/20 17/20 [ ] Draw the state transition diagram and denote all the state transition probabilities on the same. Find P[X1 = 2] List the pairs of communicating states. Find P[X2 = 3| X1 = 2] Compute P[X2 = 2 | X0 = 1] Compute P[X3 = 3, X2 = 1, X1 = 2 | X0 = 3] (vii) Find P[X4 = 4, X3 = 3, X2 = 3, X1 = 1, X0 =2] where Xt denotes the state of the random process at time instant t. The initial probability distribution is given by X0 = [2/5 1/5 1/5 1/5].
Suppose that you have 13 coins: 3 dimes, 4 quarters, and 6 nickels. Two coins are selected at random. What is the probability that you choose exactly 30 cents?

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