2. Consider the one sided random walk with states 1, 2, · ... Assume that the Markov chainhas a probability of 0.7 of moving one step to the right, and a probability of 0.3 of moving1 step to the left (P1,1 = 0.3), find the probability that the system eventually returns to1, given that it starts at 1.

probability that the system is eventually returns to 1 =3/7

Answered: 2. Consider the one sided random walk…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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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4. Consider an aperiodic, irreducible Markov chain Xn (n = 0, 1, ...) with the N + 1 states 0, 1,..., N having stationary probabilities π¿ (i = 0, 1, ..., N). Suppose that the initial distribution of the Markov Chain is chosen according to the stationary probabilities. That is, P(X = i) = πi, i = 0, 1, ..., N. Define : Qij = P(X0 = j\X₁ = i), Q = [Iij](N+1)×(N+1), 0 ≤ i, j ≤ N. (2) The square of the matrix Q is written as Q² = [93] (N+1)x(N+1). Prove that q¡ = P(X0 = j|X2 = i). dj
3. A discrete Markov model has state space equal to E {0, 1, 2}. The transition probabilities p are independent of r and recorded in the matrix P defined below. .7 .2 .1 P = .1 .6 .3 1 Assume that Lynn is in state 0 at time 0, and that i = .04. (a.) Determine the probability that Lynn is in state 0 at time 3. (b.) Determine the probability that Lynn transitions from state 1 at time 3 to state 2 at time 4.
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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.
[ 0.2 0.4 0.4 P(YIX) = |0.25 0.5 0.25| 0.4 0.3 0.3. 3- Draw the markov state diagram for the channel transition matrix shown beside.
Consider a Markov chain in the set {1, 2, 3} with transition probabilities p12 = p23 = p31 = p, p13 = p32 = p21 = q = 1 − p, where 0 < p < 1. Determine whether the Markov chain is reversible.
Given the transition matrix of a Markov process, what is the probability of going to state 3 from state 1 after 3 steps?
2. Consider a Markov chain on states {0, 1, 2, 3, 4,...} with the transition probability matrix 0 1 2 3 4 0 1 — Po Po 0 0 0 1 1 — P1 0 P1 0 0 2 1 P2 : 0 0 P2 0 ... : : Determine if the Markov chain in each case below is transient or recurrent. (i) Pn = e−1/(n+1) for n = 0, 1, 2, … · ·; (ii) Pn = e−1/(n+1)² for n = 0, 1, 2, . · ·
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A state vector X for a four-state Markov chain is such that the system is three times as likely to be in state 4 as in 3, is not in state 2, and is in state 1 with probability 0.2. Find the state vector X.

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