A Markov chain {Xn, n>0} with states 0, 1, 2, has the transition probability matrix If P{X = 0} = P{X₁ = 1} = 1, find E[X3]. 1623 H
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Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.Very very grateful!Please do not rely too much on chatgpt, because its answer may be wrong. Please consider it carefully and give your own answer. You can borrow ideas from gpt, but please do not believe its answer.
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![A Markov chain {Xn, n>0} with states 0, 1, 2, has the transition probability matrix
If P{X = 0} = P{X₁ = 1} = 1, find E[X3].
1623
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- Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3. Find limn→∞ P(Xn = 1).Q5 Suppose {Xn,n 20} is a three-state Markov chain (with states 1,2,3) with the transition probability matrix 1/3 1/3 1/3 1/2 1/2 1 . Assume that P(Xo = 1) = P(X, = 2) = 1/2. Q5 (i.) Find P(Xo = 1, X1 = 2, X2 = 3) %3D Q5 (ii.) Find P(X2 = i) for i = 1,2, 3. Q5 (iii.) Find P(T3 = 2), where T3 = inf{n > 0, X, = 3} Hint: Look at all possible ways the Markov chain reaches 'state 3' in 2-steps!!A Markov chain X₁, X₁, X₂... on states 0, 1, 2 has the transition probability matrix P (shown on the right) and initial distribution P(X₁ = 0) = 0.3, P(X。 = 1) = 0.4 and P(Xo 2) = 0.3. (1) Show that P is regular. (2) Find P(Xo = 2, X₁ = 0, X₂ = 1). (3) Find P(X₂ = 2, X₁ = 1|X₁ = 0). (4) Determine the limiting distribution. = 0 0 0.7 P = 1 0 20.5 1 2 0.2 0.1|| 0.6 0.4 0 0.5
- Consider a time-homogeneous markov chain (Xt: t = 0,1, 2, ...) with states (1,2,3}. what is P[X1 a, X4 = d | XO = i0]?Find the n-step transition probabilities pij (n) for the chain X having transition matrix P= 0 1/3 23 1211414 125212Q5 Suppose {Xn,n > 0} is a three-state Markov chain (with states 1,2,3) with the transition probability matrix (1/3 1/3 1/3 1/2 1/2 1 . Assume that P(X, = 1) = P(Xo = 2) = 1/2. Q5(i.) Find P(X, = 1, X1 = 2, X, = 3) Q5 (ii.) Find P(X, = i) for i = 1, 2, 3. Q5 (iii.) Find P(T3 = 2), where T3 = inf{n > 0, X, = 3} Hint: Look at all possible ways the Markov chain reaches 'state 3' in 2-steps!!
- 3.4 Consider a Markov chain with transition matrix 1- a a P = 1-b b 1-c where 0< a, b, c < 1. Find the stationary distribution.Suppose that the 2004 state of land use in a city of 60 mi of built-up area is C: Commercially Used 25% I: Industrially Used 20% R: Residentially Used 55%. Find the states in 2009, 2014, and 2019, assuming that the transition probabilities for 5-year intervals are given by the matrix A and remain practically the same over the time considered. From C From I From R 0.7 0.1 Tо С A =| 0.2 0.9 0.2 To I 0.1 0.8 To R A is a stochastic matrix, that is, a square matrix with all entries nonnegative and all column sums equal to 1. a Markov process.' that is, a process for which the probability of entering a certain state dep.us only on the last state occupied (and the matrix A), not on any earlier state.= 7. Let X be a continuous-time Markov chain with transition probabilities pij (t) and define F; = inf{t > T₁: X(t) =i} where T₁ is the time of the first jump of X. Show that, if gii #0, then P(F; <∞ | X (0) = i) = 1 if and only if i is persistent.
- 2.2 Let Xo. X1,... be a Markov chain with transition matrix 1 2 3 1 1/2 1/2) 1 3(1/3 1/3 1/3) and initial distribution a = (1/2,0, 1/2). Find the following: (a) P(X, = 1|X¡ = 3) (b) P(X1 = 3, X2 = 1) (c) P(X1 = 3|X2 = 1) (d) P(X9 = 1|X1 = 3, X4 = 1,X7 = 2) %3DLet P = (0.9 0.9 0.1 0.2 0.8 with two states A and B. be a transition matrix for a Markov Chain 1. What proportion of the state A population will be in state B after two steps Number 2. What proportion of the state B population will be in state B after two steps Number 3. Find the steady state vector x x1= Number X2= Number Write the results accurate to the 3rd decimal place2. Let {Xt: t = 0, 1, 2,...} be a discrete-time Markov chain. Prove that given X₂ = i, Xn+1 is independent of {Xo, X₁,..., Xn-1}. In other words, prove that P(Xn+1 = J, Xo = 20, X₁ = 1,..., Xn-1 = in-1|Xn = i) = P(Xn+1 =jXn = i) P(Xo = io, X1 = 1,..., Xn-1 = in-1 Xn = i) for any j, io, i1,..., in-1,i in the state space.