Q#5. Markov chains You may use computational software for calculations, but express your answers using proper...
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Q#5. Markov chains You may use computational software for calculations, but express your answers using proper mathematical notation. Let x₁, t € {1,2,...), be a homogenous Markov chain taking states X € {1,2,...,5} with one-step transitional probability matrix and initial distribution P = respectively. (a) Calculate var(x,). 1/10 3/10 0 0 5 5 5 ½ ¼ ¼ ¼ ½ P(0) = 6 3 6 6 1/8 1/8 1/2 ½ 01/31/01/2 (b) Calculate a stationary distribution T. 2/5 ½ ½ 0 0 0 0 0 -1 3/8 1/8 4 0 Let Y, t≥ 0, be a homogenous Markov chain taking states Y {1,2,...,5} with generator matrix (d) Let Y= = 3 and let the waiting time What is E[T3]? 1 A = 1 1/2 -31/5/ 4 1 -1 % 4 ½ ¼ ½ 6 6 0 0 0 0 0 (c) What is the probability of the state change 3-4 when the Markov chain jumps? T₁ = min(s>0|Y4+53). ½ 13/10 (e) A Markov chain is ergodic if the limit of the state probability vector, lim p(t), exists and does not depend on p(t). Using this criterion, determine if Y, ergodic? t-00 Q#5. Markov chains You may use computational software for calculations, but express your answers using proper mathematical notation. Let X₁, t = {1,2,...), be a homogenous Markov chain taking states X € {1,2,...,5} with one-step transitional probability matrix and initial distribution P = respectively. (a) Calculate var(x,). 1/10 3/10 0 0 5 5 5 ½ ¼ ¼ ¼ ½ P(0) = 6 3 6 6 1/8 1/8 1/2 ½ 01/31/01/2 (b) Calculate a stationary distribution T. 2/5 ½ ½ 0 0 0 0 0 -1 3/8 1/8 4 0 Let Y, t≥0, be a homogenous Markov chain taking states Y {1,2,...,5} with generator matrix (d) Let Y= = 3 and let the waiting time What is E[T3]? 1 A = 1 1/2 -31/5/ 4 1 -1 % 4 ½ ¼ ½ 6 6 0 0 0 0 0 (c) What is the probability of the state change 3-4 when the Markov chain jumps? T₁ = min(s>0|Y4+53). ½ 13/10 (e) A Markov chain is ergodic if the limit of the state probability vector, lim p(t), exists and does not depend on p(t). Using this criterion, determine if Y, ergodic? t-00 Q#5. Markov chains You may use computational software for calculations, but express your answers using proper mathematical notation. Let x₁, t € {1,2,...), be a homogenous Markov chain taking states X € {1,2,...,5} with one-step transitional probability matrix and initial distribution P = respectively. (a) Calculate var(x,). 1/10 3/10 0 0 5 5 5 ½ ¼ ¼ ¼ ½ P(0) = 6 3 6 6 1/8 1/8 1/2 ½ 01/31/01/2 (b) Calculate a stationary distribution T. 2/5 ½ ½ 0 0 0 0 0 -1 3/8 1/8 4 0 Let Y, t≥ 0, be a homogenous Markov chain taking states Y {1,2,...,5} with generator matrix (d) Let Y= = 3 and let the waiting time What is E[T3]? 1 A = 1 1/2 -31/5/ 4 1 -1 % 4 ½ ¼ ½ 6 6 0 0 0 0 0 (c) What is the probability of the state change 3-4 when the Markov chain jumps? T₁ = min(s>0|Y4+53). ½ 13/10 (e) A Markov chain is ergodic if the limit of the state probability vector, lim p(t), exists and does not depend on p(t). Using this criterion, determine if Y, ergodic? t-00 Q#5. Markov chains You may use computational software for calculations, but express your answers using proper mathematical notation. Let X₁, t = {1,2,...), be a homogenous Markov chain taking states X € {1,2,...,5} with one-step transitional probability matrix and initial distribution P = respectively. (a) Calculate var(x,). 1/10 3/10 0 0 5 5 5 ½ ¼ ¼ ¼ ½ P(0) = 6 3 6 6 1/8 1/8 1/2 ½ 01/31/01/2 (b) Calculate a stationary distribution T. 2/5 ½ ½ 0 0 0 0 0 -1 3/8 1/8 4 0 Let Y, t≥0, be a homogenous Markov chain taking states Y {1,2,...,5} with generator matrix (d) Let Y= = 3 and let the waiting time What is E[T3]? 1 A = 1 1/2 -31/5/ 4 1 -1 % 4 ½ ¼ ½ 6 6 0 0 0 0 0 (c) What is the probability of the state change 3-4 when the Markov chain jumps? T₁ = min(s>0|Y4+53). ½ 13/10 (e) A Markov chain is ergodic if the limit of the state probability vector, lim p(t), exists and does not depend on p(t). Using this criterion, determine if Y, ergodic? t-00
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a Calculate varXt To calculate the variance of the Markov chainXtwe can use the following equation varXt EXt EXt2 We know that the expected value of t... View the full answer
Related Book For
Managerial Economics
ISBN: 978-0133020267
7th edition
Authors: Paul Keat, Philip K Young, Steve Erfle
Posted Date:
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