Consider a sequece of iid random variables {§n, n = 0, 1, 2, . . .} with mass probabilitiesP(§n = 0) = 0.3, P(§n = 1) = 0.4, and P(§n = 2) = 0.3.Define a Markov chain (X₂)n≥o on the state space S = {0, 1, 2} using the rule Xn = min(Xn-1, §n).Write the transition matrix for this Markov chain:P =

A transition matrix is a mathematical representation used in Markov chains and other stochastic…

Answered: Consider a sequece of iid random…

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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Which of the following terms best describes the Markov property? O finiteness O memorylessness O symmetry the long run « Previous
11. A certain mobile phone app is becoming popular in a large population. Every week 10% of those who are not using the app, either because they don't have it yet or have it but are not using it, start using it, and 15% of those who are using it stop using it. Assume that the starting percentages are that 80% are not using it and 20% are using it. (a) Show the Markov matrix A representing the situation. (Letting .80 represent the starting 1.20 .80 percentages, remember that you want the situation after one week to be given by A .) .20 (b) What percentage will be using the app after two weeks? (c) Find the eigenvalues and eigenvectors of A. (d) Show a matrix X which diagonalizes A by means of X-'AX. (d) In the long run the percentages not using the app will be Show your work. _% and using it will be _%
Suppose that a production process changes state according to a Markov chain on [25] state space S = {0, 1, 2, 3} whose transition probability matrix is given by a) Determine the limiting distribution for the process. b) Suppose that states 0 and 1 are “in-control,” while states 2 and 3 are deemed “out-of-control.” In the long run, what fraction of time is the process out-of-control?
Suppose I flip n independent biased coins such that the jth coin has probability j/n of being heads, for j = 1,...,n. Let Hn be a random variable equal to the total number of heads. (a) What is the expectation of H,? (b) What is the variance of Hn? (c) Use Markov's inequality to derive an upper bound on P(Hn 2 9n/10).
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
Find the limiting distribution for this Markov chain.
Q. No. 42 Let {X,}nzo be a homogeneous Markov chain whose state space is {0,1,2} and 1 whose one-step transition probability matrix is P = 0.3 0.7. Then 1 lim P(X2n = 2 |X, = 2) = (correct up to one decimal place). n-00
Consider a Markov chain in the set {1, 2, 3} with transition probabilities p12 = p23 = p31 = p, p13 = p32 = p21 = q = 1 − p, where 0 < p < 1. Determine whether the Markov chain is reversible.
= 3. Propose an algorithm for simulating the occurrence of two dependent events A and B that uses only one uniform random variable U~Unif(0, 1). Assume that the probabilities P(A) ΡΑ, P(B) = PB, and P(AUB) = q are given, where PA, PB, q = (0, 1) and max(PA, PB) < q < PA+PB· Formulate your answer in the form of the following statement: "if U E ..., then event A occurs; if U € .……., then event B occurs."
Determine the probability transition matrix ∥Pij∥ for the following Markov chains. Pleaseprovide the solutions step by step and provide a short explanation for each step. Let the discrete random variables ξ1, ξ2, . . . be independent and with thecommon probability mass function
9. |Show that if X,, X2,... is a Markov chain, then it is statistically determined by its |second-order probability masses: P{X, = Xị , X, = x}, for any i, j = 1, 2,..
Q3) The state transition matrix of a Markov random process is given by 1/3 1/3 1/6 1/6 5/9 0 0 4/9 2/5 1/5 1/5 1/5 0 0 3/20 17/20 [ ] Draw the state transition diagram and denote all the state transition probabilities on the same. Find P[X1 = 2] List the pairs of communicating states. Find P[X2 = 3| X1 = 2] Compute P[X2 = 2 | X0 = 1] Compute P[X3 = 3, X2 = 1, X1 = 2 | X0 = 3] (vii) Find P[X4 = 4, X3 = 3, X2 = 3, X1 = 1, X0 =2] where Xt denotes the state of the random process at time instant t. The initial probability distribution is given by X0 = [2/5 1/5 1/5 1/5].

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