1. True/False. For each of the following statements, write T (True) if the statement is necessarily true, F (False) if the statement could possibly be false, or U (unsure) if you are unsure of the answer. "You do not need to explain your answer" (a) If a Markov chain is transient (i.e. all states are transient), then it has no invariant measure. (b) A recurrent Markov chain may not have an invariant distribution. (c) If a Markov chain (Xn)nzo is irreducible and positive recurrent, that the distribution of Xn will converge to certain limiting distribution as n → ∞. (d) Let (Xk)k>1 be an i.i.d. sequence with distribution 1 1 P(X₁ = −2) = 2, P(X₁ = 0) = P(X₁ = 4) = and set W₁ = X₁ +...+ X₂. Then the sequence (Wn)n>1 is a martingale with respect to (Xk)k>1. (e) For the symmetric simple random walk (Sn)nzo on Z and any stopping time T with P(T < ∞) = 1, we always have E[ST] = E[So].

1. True/False. For each of the following statements, write T (True) if the statement is necessarily true, F (False) if the statement could possibly be false, or U (unsure) if you are unsure of the answer. "You do not need to explain your answer" (a) If a Markov chain is transient (i.e. all states are transient), then it has no invariant measure. (b) A recurrent Markov chain may not have an invariant distribution. (c) If a Markov chain (Xn)nzo is irreducible and positive recurrent, that the distribution of Xn will converge to certain limiting distribution as n → ∞. (d) Let (Xk)k>1 be an i.i.d. sequence with distribution 1 1 P(X₁ = −2) = 2, P(X₁ = 0) = P(X₁ = 4) = and set W₁ = X₁ +...+ X₂. Then the sequence (Wn)n>1 is a martingale with respect to (Xk)k>1. (e) For the symmetric simple random walk (Sn)nzo on Z and any stopping time T with P(T < ∞) = 1, we always have E[ST] = E[So].

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 9EQ

See similar textbooks