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.

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...

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn ≤ lim inf(xn + yn) ≤ lim sup(xn + yn) ≤ lim sup xn + lim sup yn. Give example where all inequalities are strict. b)Let (zn) be the sequence defined recursively by z1 = z2 = 1, zn+2 = √ zn+1 + √ zn, n = 1, 2, . . . . Prove that (zn) is convergent and find its limit. Hint; argue...

1. . Let X1, . . . , Xn, Y1, . . . , Yn be...

1. . Let X1, . . . , Xn, Y1, . . . , Yn be mutually independent random variables, and Z = 1 n Pn i=1 XiYi . Suppose for each i ∈ {1, . . . , n}, Xi ∼ Bernoulli(p), Yi ∼ Binomial(n, p). What is Var[Z]? 2. There is a fair coin and a biased coin that flips heads with probability 1/4. You randomly pick one of the coins and flip it until you get a...

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...

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...

Give a counterexample: a) Xn + Yn converges if and only if both Xn and Yn...

Give a counterexample: a) Xn + Yn converges if and only if both Xn and Yn converge. b) Xn Yn converges if and only if both Xn and Yn converge.

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.

6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M =...

6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M = max(X1,...,Xn). (a) Find the density function of M. (b) Find E[M] and V[M].

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