STATS 325721 Assignment

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STATS 325/721 Assignment 21. Consider a Markov chain X = (Xn : n = 0, 1, 2, . . .) on the set of vertices {A, B, C, D, E}of the 3D object with six triangular faces represented in the following diagram:Let Pi and Ei represent probability and expectation, respectively, conditional onX0 = i. The first hitting time of state i is defined byTi:= inf{n > 0 : Xn = i}.(a) Prove thatPB(TA < TD) = 37,where you should justify your steps. [4](b) Calculate PE(TB < TA; TB < TD), and deduce PC(TB < TD < TA). [4](c) Find the average time, EA(TB), it takes to reach state B starting initially instate A. [4](d) Deduce the average time to return to state B starting from state B. Deducethe long term proportion of time spent in each state. [∗3]2. Consider the Markov chain X = (Xn)n∈N with state space I = {A, B, C, D, E, F}and one step transition probabilities given in the following diagram:(a) Decompose the state space into its communicating classes and state the period

代写STATS 325/721作业、代做R编程设计作业、R课程设计作业代写of each class. Hence, identify the set of transient states T and a communicatingclass of recurrent states R. [3]Due noon, Monday, September 2nd 2019 (Science SRC)STATS 325/721 Assignment 2(b) Write down the one-step transition matrix P for the discrete parameter Markovchain Y with state space R, that is, the restriction of the Markov chain X tothe recurrent class R ⊂ I. [3](c) What conditions does an invariant probability mass function π for a discretetime Markov chain satisfy? Find π for the Markov chain Y . [3](d) Stating any general results that you appeal to, deduce the following:i. Y is positive recurrent, [1]ii. the distribution for the position of Y after the chain has been running fora very long time, [1]iii. the long-term proportion of time spent in each of the states, [1]iv. the average time to return to each state EiTi, [1]v. the average number of visits made to A before returning to the startingposition at C. [∗2]3. Consider the random walk W = (Wn)n≥0 with state space Z such thatWn := W0 + X1 + · · · + Xn,where X1, X2, . . . are independent, identically distributed random variables withP(Xn = −1) = 25, P(Xn = 1) = 15, P(Xn = 2) = 25.(a) For k ≥ 1, let xk be the probability that the random walk ever visits the origingiven that it starts at position k, that is,xk := Pk(hit 0) := P(Wn = 0 for some n ≥ 0 | W0 = k).i. By splitting according to the first move, show thatand explain why xk = (x1)kfor k ≥ 1. [5]ii. Show that Pk(hit 0) = 2−kfor k ≥ 1. [5](b) For k ≥ −1, let yk be the probability that the random walk ever visits k giventhat it starts at 0, that is,yk := P0(hit k) := P(Wn = k for some n ≥ 0 | W0 = 0).i. Write down the values of y−1 and y0. [2]ii. For k ≥ 1, briefly explain whyiii. Find all solutions to (∗) of the form yk ∝ mk and write down the generalsolution of the recurrence relation (∗). Deduce P0(hit k) for k ≥ −1. [∗4]iv. If the random walk starts at the origin and n > 0 is a very large integer, deducethat the probability that position n is never visited is approximately1/6. [∗1]Stats325: mark is out of 40, including max 5 bonus from ∗starred questionsStats721: mark is out of 50, please attempt all questionsDue noon, Monday, September 2nd 2019 (Science SRC)

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