The transition matrix of a Markov chain is given by0 0P =(a) Find two distinct stationary distributions of this Markov chain.(b) Find the general form of the stationary distribution.(c) If 70) = (, 4, ¿, 1, ) is the initial probability vector at time 0,,π(m) = (금' 등,을 ,).4'4'6' 6' 6then show that limn12HIN O -160 2/3HIN O 230116O 0 -16 0 116

Given the transition matrix of the Markov chain as P=0 0 12 0 12034014016023016016056016016023

Answered: The transition matrix of a Markov chain…

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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• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pu(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1 : Xn = 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. 1. True/False. For each of the following statements, write T (True) if the statement is necessarily true, F (False) if the statement could possibly be false, or U (unsure) if you are unsure of the answer. "You do not need to explain your answer" (a) If a Markov chain is transient (i.e. all states are transient), then it has no invariant measure. (b) A recurrent Markov chain may not have an invariant distribution. (c) If a Markov chain (Xn)nzo is irreducible and positive recurrent, that the distribution of Xn will converge to certain limiting distribution as n → ∞. (d) Let (Xk)k≥1 be an i.i.d. sequence with distribution P(X₁ = = −2) = 1 P(X₁ =…
6. Which of the following (when stationary) are reversible Markov chains? (₁ "B) (a) The chain X = {X₂} having transition matrix P = (b) The chain Y = {Yn} having transition matrix P = 1 - a ( В 0 P 0 Р where a + ß> 0. P P P 1-p 0 (c) Zn = (Xn, Yn), where Xn and Yn are independent and satisfy (a) and (b). where 0 < p < 1.
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8/ Example 7-36. Let {Xg} be mutually independent, each assuming the values 0,1.2 3 ..., a – 1 with probability Let S„ = X1 + X2 + ... + Xp. Show that the probability generating P (s) = { 6)= { 1-s a(1 – s) function of S, is : P (S„ = j) =£ (-1)**jtev (") (;="av) and hence V=0
2.- Let X₁, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix 1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 0 0 2/3 1/6 1/6, (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it. Your answer should refer to the relevant Theorems in the notes. (b) Find the limiting distribution for this Markov chain. (100) (c) Without doing any more calculations, what can you say about p₁,1 (100) ? and P2,1
Let X be a point randomly chosen on a stick of length 1 with a uniform distribution. Let Y be a point chosen between 0 and X on this stick with a uniform distribution. Y X 1 Find the covariance between X and Y. Please provide the solution step by step.
4. Suppose Xn is a two-state Markov chain whose transition probability matrix P is 0 1 0 Q 1-a 1 1-8 B Then, Zn (Xn-1, Xn) is a Markov chain having the four states (0, 0), (0, 1), (1,0), and (1,1). Determine the transition probability matrix.
3. Let X ~ Exponential() and let t be a constant with 0 0 be any value. (a) Calculate P(X > b) exactly. (b) Now suppose we didn't know the answer to (a), i.e., we could not calculate P(X > b) exactly. Recalling that E(X)= , use Markov's inequality to establish an upper bound on P(X > b). (c) Next we'll apply Markov's inequality in a different way. First, calculate E(etx). (d) Next, use the Markov inequality to prove a bound on P(etX > a) (here a > 0 is any positive number, while we assume 0 b) (here b > 0 is any positive number, and again 0 b) as in part (a), the upper bound on P(X > b) as in part (b), and the alternative upper bound on P(X > b) as in part (e). How close are these upper bounds to the real probability? Which bound is better? Try this for a few values of X, b,t and describe what you see. 6.
Suppose that a Markov chain (X,)n>o has a stochastic matrix given by: 1/2 1/2 1/4 3/4 1/3 1/3 1/3 P = 1/6 2/3 1/6 2/3 1/3 1/6 5/6 1/2 0 0 0 1/2 (c) What is P4(H' < H')? (d) What is P2(H' < H')? (e) Suppose now that the Markov chain is initialised from 1, which is a uniform distribution over all states 1 to 7. What is Pa(H' < H')? (f) What is the probability that a Markov chain, which started in state 2, visits state 1 exactly once before hitting state 7?
3. Refer to the random walk on the right. a) Find a transition matrix A so that p(n + 1) = A p'(n). b) Assuming you start at node 2, compute p(0), p (1), p(2), p'(3), p'(4). 0.3 H 0.7 0.7 0.3 c) Solve A .x = x. There should be infinitely many solutions. Find the one of them which equals lim p'(n). n→∞
- We assume that the stochastic process for a stock price is an Arithmetic Brownian motion, with a drift of 49% and, diffusion of 39%. Find the probability that the stock price will be between 0.17 and 1.32 in 6 years. A) 0.06 (B) 0.05 (C) 0.04 (D) 0.03 (E) 0.02 Select v Save
A Markov chain has the transition probability matrix 0.3 0.2 0.5 0.5 0.1 0.4 [0.5 0.2 0.3 Given the initial probabilities o1 = ¢2 = 0.2 and ø3 0.6, what is Pr (Xı = 3, X2 = 1)? %3D

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