(a) (Xn+r)nzo for fixed r≥ 0,
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![Let (Xn)nzo be a Markov chain. Show that the following are also Markov chains:
(a) (Xn+r)n≥o for fixed r ≥ 0,
(b) (X2n)n≥0,
(c) the sequence of pairs ((Xn, Xn+1))n>0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f1bdb73-a19f-4e46-b05e-06d52386dad5%2F23770516-b707-43c6-8871-304849a37df1%2Fxmsa11w_processed.png&w=3840&q=75)
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- (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?• 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. 2. A bank classifies loans as paid in full(F), in good standing (G), in arrears (A), or as a bad debt (B). Loans move between the categories according to the transition matrix: P = F G A B F 1 0 0 0 G .1 .8 .1 0 A .1 .4 .4 .1 B 0 0 0 1 What fraction of loans in good standing are eventually paid in full? (That is, starting at state G, what is the probability that the chain eventually visits F?)• 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 : Xn = 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. 3. Each year an auto insurance company classifies its customers into three categories: Poor, Satisfactory, and Good. No one moves from poor to good or from good to poor in one year. The status of a driver can be modeled by a Markov chain {Xn : n ≥ 0} with state space S = {P, S, G} and transition matrix PS G P/1/2 1/2 0 S 1/5 3/5 1/5 G 0 1/5 4/5 (a) Compute the invariant distribution for this Markov chain. (b) Assume that the average prices (per year) of insurance policies for drivers with Poor, Satisfactory, and Good ratings are $600, $300, $200, respectively. In the long run, how much does a driver pay for the insurance per year? (c) For a…
- 4. Suppose Xn is a two-state Markov chain whose transition probability matrix P is 0 1 0 Q 1-a 1 1-8 B Then, Zn (Xn-1, Xn) is a Markov chain having the four states (0, 0), (0, 1), (1,0), and (1,1). Determine the transition probability matrix.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]?For each of the following Markov chains, determine the long run fraction of the time that each state will be occupied. 1 3 3 (a) 1 1 - 2 2 0.8 0.2 (b) 0.2 0.8 0.8 0.2 A) 0.6 and 0.4, 0.64,0.2 and 0.16 В) 0.3 and 0.7, 0.64,0.2 and 0.16 C) 0.6 and 0.4, 0.55,0.25 and 0.15 D) None of the answer choices are correct.
- A Markov chain model for a species has four states: State 0 (Lower Risk), State 1 (Vulnerable), State 2 (Threatened), and State 3 (Extinct). For t 2 0, you are given that: 01 zit = 0.03 12 t= 0.05 23 Hit = 0.06 This species is currently in state 0. Calculate the probability this species will be in state 2 ten years later. Assume that reentry is not possible. (Note: This question is similar to #46.2 but with constant forces of mortality) Possīble Answers A 0.02 0.03 0.04 D 0.05 E 0.06Help me fast so that I will give Upvote.Determine whether the statement below is true or false. Justify the answer. If (x) is a Markov chain, then X₁+1 must depend only on the transition matrix and xn- Choose the correct answer below. O A. The statement is false because x, depends on X₁+1 and the transition matrix. B. The statement is true because it is part of the definition of a Markov chain. C. The statement is false because X₁ +1 can also depend on X-1 D. The statement is false because X₁ + 1 can also depend on any previous entry in the chain.
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