Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1,...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3} and transition probability matrix (pij ). Let τi = min{n ≥ 1 : Xn = i}, i = 0, 1, 2, 3. Define Bij = {Xτj = i}. Is Bij ∈ σ(X0, · · · , Xτj ) ? Q2. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, X0 = 0, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Let Bij as in Q1. Compute P(Xτ0+2 = 2|B00).

Q4. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2} and transition probability matrix (pij ) given by   2 3 1 3 0 1 3 2 3 0 0 1 4 3 4   Let τ2 = inf{n ≥ 1 : Xn = 2} Find P{τ2 < ∞|X0 = 2}. Q5. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Determine all recurrent states. 1

Expert Solution

orchestra answered 1 year ago

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S. For i ∈...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S. For i ∈ S, define τi = min{n ≥ 0|Xn = i}. Show that τi is a stopping time for each i. Q2. Let τi as in Q1. but for any discrete time stochastic process. Is τi a stopping time? Q3. Let {Xn|n ≥ 0} be a Markov chain and i is a state. Define the random time τ = min{n ≥ 0|Xn+1 = i}. If τ...

Q5. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1,...

Q5. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Determine all recurrent states. 1 2 Q6. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2} and transition...

Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2,...

Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Determine all recurrent states. Q3. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2} and transition probability matrix (pij...

Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · ·...

Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · · · } and transition matrix (pij ) given by pij = 1 2 if j = i − 1 1 2 if j = i + 1, i ≥ 1, and p00 = p01 = 1 2 . Find P{X0 ≤ X1 ≤ · · · ≤ Xn|X0 = i}, i ≥ 0 . Q2. Consider the Markov chain given in Q1. Find P{X1,...

1.14 Let Xn be a Markov chain on state space {1,2,3,4,5} with transition matrix P= 0...

1.14 Let Xn be a Markov chain on state space {1,2,3,4,5} with transition matrix P= 0 1/2 1/2 0 0 0 0 0 1/5 4/5 0 0 0 2/5 3/5 1 0 0 0 0 1/2 0 0 0 1/2 (a) Is this chain irreducible? Is it aperiodic? (b) Find the stationary probability vector.

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4...

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4 0.2 0.4 P = 0.6 0.3 0.1 0.5 0.3 0.2 And initial probability vector a = [0.2, 0.3, 0.5] For the Markov Chain with state-space, initial vector, and transition matrix discuss how we would calculate the follow; explain in words how to calculate the question below. a) P(X1 = 0, X2 = 0, X3 = 1, X4 = 2|X0 = 2) b) P(X2 =...

Let Xn be the Markov chain with states S = {1, 2, 3, 4} and transition...

Let Xn be the Markov chain with states S = {1, 2, 3, 4} and transition matrix. 1/3 2/3 0 0 2/3 0 1/3 0 1/3 1/3 0 1/3 0 1/3 2/3 0 a.) Let X0 = 3 and let T3 be the first time that the Markov chain returns 3, compute P(T3 = 2 given X0=3). Please show all work and all steps. b.) Find the stationary distribution π. Please show all work and all steps.

Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7...

Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7 .3 .3 .4 .6 .3 .1 a) find E[X1|X0=2] = b) The P(X9=1|X7=3) = C) The P(X2=2) =

Q1. Let {Xn : n ≥ 0} denote the random walk on 9-cycle. Express it as...

Q1. Let {Xn : n ≥ 0} denote the random walk on 9-cycle. Express it as a random walk on a group (G, ·) with transition probabilities given by pxy = µ(y · x −1 ) for an appropriate distribution µ on G. Q2. Consider the stochastic process {Xn|n ≥ 0}given by X0 = 1, Xn+1 = I{Xn = 1}Un+1 + I{Xn 6= 1}Vn+1, n ≥ 0, where {(Un, Vn)|n ≥ 1} is an i.i.d. sequence of random variables such...

Let X = {Xn:n= 0,1,...,} be a DTMC on state space S. Define Yn = (Xn,...

Let X = {Xn:n= 0,1,...,} be a DTMC on state space S. Define Yn = (Xn, Xn+1). Prove that Y = {Yn: n = 0,1,2,...} is a DTMC. Specify its state space and the transition matrix.

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