Q. Let us consider that, {X,} is a Markov chain with 5 states 0, 1, 2, 3 and 4. Suppose, Po4 =1 and when the state is in the state i, i > Othe next isequally likely to be in any of the states 0, 1, 2, .., i - 1a. What is the nature of the states of this Markov chain.b. Find whether there exists a limiting distribution of this this Markov chain and compute that if it exists.

Given that Xn is a Markov chain with 5 states 0, 1, 2, 3 and 4.

Answered: Q. Let us consider that, {X,} is a…

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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9. |Show that if X,, X2,... is a Markov chain, then it is statistically determined by its |second-order probability masses: P{X, = Xị , X, = x}, for any i, j = 1, 2,..
Suppose you have 52 cards with the letters A, B, C, ..., Z and a, b, c, ..., z written on them. Consider the following Markov chain on the set S of all permutations of the 52 cards. Start with any fixed arrangement of the cards and at each step, choose a letter at random and interchange the cards with the corresponding capital and small letters. For example if the letter "M/m" is chosen, then the cards "M" and "m" are interchanged. This process is repeated again and again. (a) Markov chain. Count, with justification, the number of communicating classes of this (b) Give a stationary distribution for the chain. (c) Is the stationary distribution unique? Justify your answer. (d) initial state. Find the expected number of steps for the Markov chain to return to its
60. The following is the transition probability matrix of a Markov chain with states 1, 2, 3, 4 P= .4 .2 .25 .2 .3 .2 .1 .2 .2 .4 25 50 .1 .4 .3 If Xo = 1 (a) find the probability that state 3 is entered before state 4; (b) find the mean number of transitions until either state 3 or state 4 is entered.
1. For each of the transition matrices given below, list absorbing state(s), and determine the period of each of the states in each Markov chain. (a) P = (d) P = 1120 CLI2L 1434L3 0 ܚ . ܟ ܝ. | ܟ ܝܕ . ܚ 0 00 0 0 0 0 0 3I4LI4 LIABI+ 0 0 (b) P = 1 0 Lo (e) P = 0 1 1 0 0 1333 COLITOLIN 0 0 0 0 0 213 100 0 0 3 00 100 0 -0 WIN 1 0 0 0 0 0 0 00 1 0 0 4|012 TWO O 0 0 0 0 0 (c) P = (f) P = 0 1|2 1|3 OLT20IN +15 O N|TN|TO 120 HINO OLI3 O 0 LI5L4L 1113 2I5
Please do the following question with handwritten working out
40. A Markov Chain with 4 states is currently equally likely to be in states 3 and 2, but is 4 times as likely to be in state 1 as in 3, and is 2 times as likely to be in 4 as in 1. What is the state vector that describes this situation?
8. List the Gauss–Markov conditions required for applying a t & F-tests.
Consider the Markov chain with three states,S={1,2,3}, that has the following transition matrix 1.Draw the state transition diagram for this chain. (10 marks)
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].
2. For all permissible p values, determine the equivalence classes of the Markov chain with the following transition matrix P, classify states as transient or recurrent, and classify the Markov chain as irreducible or reducible. 0. 1-p 1 - 0. P = 0. 1-p 0. 1-p
1. Prove that Weiner Process is Markovian (Markov Process).
2. Let Zbe a Markov chain on a state space S = {1,… … , m} with transition probability matrix P(i, j) = Tij. Suppose that sample paths of Z are observed from which we record Njj #i →→ j transitions of Z, m N₁ = [N₁; (# transitions out of state i). j=1 It is known that the likelihood function of the observations of Z is given by m Nij ΠΠΠ. i=1_ j=1 L = Incorporating the constraint j = 1 for each i = S, j=1 m (a) Write the loglikelihood function of the observation. (b) Show that the maximum likelihood estimator of i → j transition probability ij is N = Tij = (c) Let m = 2 and assume that the number of transitions of Z is given by the matrix Nij N₁ 50 100 25 75 Use the above information to derive estimators for 11, 12, 721 and 22.

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