1. Assume X are independent and identically distributed with P(X₁ = 1) = p, P(X₁ = 0) = r and P(X₁ -1) = q. where p, r,q > 0 and p+r+ q = 1. = Let Sn=Xi, n = 1, 2, . . .. (a) Prove that {S1, S2,...} is an irreducible Markov chain with state space S = {0,±1, ±2, ...} and write down its transition matrix. (b) Is the chain aperiodic? (c) Find expressions for: i. P(S3 = 2). ii. P(S₁ = 1|S₁ = 1). iii. P(S10=1|S7 = 0). iv. ES and var(Sn).

1. Assume X are independent and identically distributed with P(X₁ = 1) = p, P(X₁ = 0) = r and P(X₁ -1) = q. where p, r,q > 0 and p+r+ q = 1. = Let Sn=Xi, n = 1, 2, . . .. (a) Prove that {S1, S2,...} is an irreducible Markov chain with state space S = {0,±1, ±2, ...} and write down its transition matrix. (b) Is the chain aperiodic? (c) Find expressions for: i. P(S3 = 2). ii. P(S₁ = 1|S₁ = 1). iii. P(S10=1|S7 = 0). iv. ES and var(Sn).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 14EQ

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