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Let Ek, k > 0, be independent random variables taking only the integers. Let S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable, independent of the sequence &k, taking values in S and let ƒ : S × Z → S be a certain real function. Define new random variables X,„ by Xn+1 = f(Xn, En), n = 0, 1,2.. (a) Show that {X„}n21 is a Markov chain with state space S = {1, ..., N} and this chain is homogeneous if only §k, k > 1, have the same distribution.

Let Ek, k > 0, be independent random variables taking only the integers. Let S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable, independent of the sequence &k, taking values in S and let ƒ : S × Z → S be a certain real function. Define new random variables X,„ by Xn+1 = f(Xn, En), n = 0, 1,2.. (a) Show that {X„}n21 is a Markov chain with state space S = {1, ..., N} and this chain is homogeneous if only §k, k > 1, have the same distribution.

MATLAB: An Introduction with Applications
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 Ek, k > 0, be independent random variables taking only the integers. Let S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable, independent of the sequence &k, taking values in S and let f : S x Z → S be a certain real function. Define new random variables X, by Xn+1 f(Xn, §n), n = 0, 1, 2... (a) Show that {X„}n21 is a Markov chain with state space S = {1, .., N} and this chain is homogeneous if only Fk, k > 1, have the same distribution. (b) Suppose that Xo = 0, f(x, y) = x+y and fk, k > 0, are iid with P({o = 1) = p, P(5o = 0) =r and P(£o = -1) = q. where p,r, q > 0 and p+r+q = 1. i. Write down the transition probabilities of the chain {Xn}n>1- ii. Is the chain {Xn}n>1 aperiodic and irreducible? iii. Find expressions for: P(X3 = 2), P(X4 = 1|X1 = 1) and P(X10 = 1|X7 = 0).
Transcribed Image Text:Let Ek, k > 0, be independent random variables taking only the integers. Let S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable, independent of the sequence &k, taking values in S and let f : S x Z → S be a certain real function. Define new random variables X, by Xn+1 f(Xn, §n), n = 0, 1, 2... (a) Show that {X„}n21 is a Markov chain with state space S = {1, .., N} and this chain is homogeneous if only Fk, k > 1, have the same distribution. (b) Suppose that Xo = 0, f(x, y) = x+y and fk, k > 0, are iid with P({o = 1) = p, P(5o = 0) =r and P(£o = -1) = q. where p,r, q > 0 and p+r+q = 1. i. Write down the transition probabilities of the chain {Xn}n>1- ii. Is the chain {Xn}n>1 aperiodic and irreducible? iii. Find expressions for: P(X3 = 2), P(X4 = 1|X1 = 1) and P(X10 = 1|X7 = 0).
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