5. Let X be a continuous-time Markov chain with generator G satisfying gi = -8ii > 0 for all i.Let HA = inf{t 20: X(t) = A} be the hitting time of the set A of states, and letn; = P(HA <∞ |X (0) = j) be the chance of ever reaching A from j. By using properties of the jump chain, whichyou may assume to be well behaved, show that Σk 8jknk = 0 for j & A.

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Answered: 5. Let X be a continuous-time Markov…

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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Let X Exponential (3) a) Find the Markov upper bound for P(X>10). b) Find the Chebyshev upper bound for P(X>10). State which of the two bounds found is better. c) Find the Chernoff upper bound for P(X>10). Clearly state the allowable range for t. d) Let Zn be a RV from a Gamma(n, 3) distribution for n=1,2,... . What does converge in probability to? Please state both the value and the reasoning by which Z converges in probability to it. n
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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
Show that if X is not a deterministic random variable, then H(X) is strictly positive. What happens to the probabilities if a random variable is non-deterministic?
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A Markov chain {x}=0,1,2,.. satisfies the difference equation X = Axk-1 for every k ≥ 1, where - (0.9 A = {) 0.8 0.6 0.2 0.4 (i) Find the general term x for k≥ 1. (ii) What happens to x as k→ ∞o? - (0.3). and Xo =
• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = 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. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1
9. Let i be a transient state of a continuous-time Markov chain X with X (0) : = i. Show that the total time spent in state i has an exponential distribution.

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