Consider three r.v.'s X1, X2, X3 with finite sample space X; E {1,..., K}. The sequence X¡, i = 1, 2, 3,is called a Markov chain ifPr (X3 = i| X2 = j, X1 = k) = Pr (X3 = i | X2 = j)= Pr (X2 = i| X1 = j) .That is, Pr (X3 = i| X2 = j) = Pi does not depend on X1, and it is the same as Pr (X2 = i| X1 = j).We call Pji the transition probabilities.Let P = [Pi] denote the (K × K) matrix of transition probabilties.Let T = (T1,..., TK)' be a probability vector (i.e., T; > 0 and Tk = 1) withTk Pkj = T;Pjk%3Dfor any pair of states j and k. If q1 = T, show that q2 = q3 = r as well (T is called an equilibriumdistribution").By the law of total probability φΣ, Τ P- Σ, ΤP πι Σ, P= πιBy Bayes' theorem q2i =92i = Ti%3Dq1 is a probability vector and P is a stochastic matrix = q2 = q3 = 91.By definition of conditional probability PjPr(X1=k,X2=j)Pr(X1=k)= Tj.%3D%3Dnone of these

We have three r.v.'s X1 X2 X3 with finite sample Space :

Answered: Consider three r.v.'s X1, X2, X3 with…

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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7. The following 4-state Markov model with constant intensities a, B and y is used to model a pension scheme for male members aged 55. Age is defined as age last birthday on January 1". The states are w: Active, a: Preserved Benefit, b: Drawing pension and d: Dead. The following data were collected over an observation period covering a single calendar year on 1000 male members of the above pension scheme, all of whom are aged 55: Total waiting time in state w: 800 years Total number of transfers from state w to state a: 200 Total number of transfers from state w to state b: 200 Total number of transfers from state w to state c: 20 Find the likelihood function L(a, B.y) for these data and hence find the maximum likelihood estimates of a,ß and y.
4. Suppose X0, X1, X2,... are iid Binomial (2, 5). If we view this sequence as a Markov chain with S = {0,1,2}, what is its PTM?
None
3.2: Markov Chains are Proper Probabilistic Models Show that the sum of the probabilities over all possible sequences of any length is 1. This proves that the Markov chain describes a proper probability distribution over the whole space of sequences.
= 3. Propose an algorithm for simulating the occurrence of two dependent events A and B that uses only one uniform random variable U~Unif(0, 1). Assume that the probabilities P(A) ΡΑ, P(B) = PB, and P(AUB) = q are given, where PA, PB, q = (0, 1) and max(PA, PB) < q < PA+PB· Formulate your answer in the form of the following statement: "if U E ..., then event A occurs; if U € .……., then event B occurs."
5. Explain what is meant by BLUE estimates and the Gauss-Markov theorem. Mathematics relevant proofs will be rewarded. [:
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
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?
prove the property
Let X be a Markov chain on S, and let I: S" {0, 1}. Show that the distribution of Xn, Xn+1,..., conditional on {I (X₁,..., Xn) = 1} n {Xn = i}, is identical to the distribution of Xn, Xn+1,... conditional on {Xn = i}.
5

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