Homework Help is Here – Start Your Trial Now!
Math
Statistics
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.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter14: Counting And Probability
Section14.2: Probability
Problem 3E: The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the...
See similar textbooks
icon
Related questions
Question
please solve all asap
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).
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
College Algebra
Algebra
ISBN:
9781337282291
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS