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MSO4311: Markov Chain - Transition Matrix - Science and Maths Assignment Help

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Added on: 2020-03-09 10:59:08
Order Code: MAS26642
Question Task Id: 95308

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Assignment Task:

Answer all questions. Marks are indicated beside each question.

1. Consider the Markov Chain {Xt}t∈N0 with the following state diagram:

Markov Chain

(a) Write down the transition matrix for this Markov Chain

(b) Find and classify the communicating classes of this Markov Chain. Write about 
• their transience/recurrence, 
• their period, 
• whether they are null or non-null (where appropriate). 

Justify your claims.

(c) What is the expected number of returns to state 10? Justify your answer.

Let Cmin be the smallest recurrent communicating class of the above Markov Chain, and consider a Markov Chain consisting of just this communicating class. 

(d) Write down the transition matrix for the Markov Chain on states Cmin and find the eigenvalues and eigenvectors for this matrix. Show your work i.e. find the eigenvalues and eigenvectors without relying on software. (You may check your answers using software). 

(e) Find a stationary distribution of the Markov Chain on states Cmin. Is this stationary distribution unique? Justify your answers.

(f) What is the mean return time to state 1? Justify your answer.

(g) Is there a limiting distribution for the Markov Chain on states Cmin? Justify your answer.

Let Cmax be the largest recurrent communicating Chain consisting of just this communicating class. class of the Markov Chain {Xt}t∈N0, and consider a Markov 

(i) Write R code to define the transition matrix of the Markov Chain on state Cmax.

(j) Write R code to diagonalise the transition matrix. The output should be a matrix M and a matrix D so that M−1DM is the transition matrix you wrote down for part (i).

(k) Consider the initial state vector 

y = (0.2 0.3 0.1 0.2 0.2). 
Find the evolution of y under the Markov Chain on states Cmax after 10-time steps.

(l) What is the limiting state of the initial state vector y? Justify your answer.

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  • Uploaded By : Alex Cerry
  • Posted on : March 09th, 2019
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