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

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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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?
4. Let X be a Markov chain with state space S = {1, 2, 3) and transition matrix (² where 0 < p < 1. Prove that P = 0 Ph = P P P 1-P 0 0 P 1-p ain a2n a3n aln a2n a3n where ain + wa2n + w²a3n = (1 − p + pw)", w being a complex cube root of 1. a3n a2n ain
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 =
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• 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
(6) Give an example of a Markov chain Xn on a countable state space S and a function g on S such that g(Xn) is not a Markov chain. Can you give any conditions on Xn and g which ensure that g(Xn) is a Markov chain?
a)
Please do the following question with handwritten working out
c) Consider the transition Markov chain with three states S = (1, 2, 3) as shown below. %3D 1 Given that; P(X, = 1) = 3' fine P(X, = 1,X, = 2,X2 = 3) N/3 1/2 1/2 1/4 1/2 1/4
In a science fair project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 352 trials, the touch therapists were correct 168 times. Complete parts (a) through (d). 20 a. Given that Emily used a coin toss to select either her right hand or her left hand, what proportion of correct responses would be expected if the touch therapists made random guesses? (Type an integer or a decimal. Do not round.) an example 5 Get more help. M B D CHL & 7 GEREED Clear all 9 UK Check answer Us Mar 1
C)

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