[SOLVED] PSTAT 160A Fall 2024 – Assignment-4

PSTAT 160A Fall 2024 – Assignment-4

(Due by November 30 (Saturday), midnight 11:59 pm.)

1. Find the communication classes of a Markov chain with transition matrix

Rewrite the transition matrix in canonical form.

2. Show that the stationary distribution for the modified Ehrenfest chain is binomial with param-eters N and 1/2.

3. Consider a Markov chain with transition matrix

Identify the communication classes. Classify the states as recurrent or transient, and determine the period of each state.

4. Markov chains are used to model nucleotide substitutions and mutations in DNA sequences. Kimura gives the following transition matrix for such a model.

Find a vector x that satisfies the detailed-balance equations. Show that the chain is reversible and find the stationary distribution.

5. Consider a Markov chain with transition matrix

where 0 < α, β, γ < 1. Find the transition matrix of the time reversal chain.

6. Classify the states of a Markov chain, with S = {0, 1, 2, 3}, given by the transition probability matrix

Write the matrix in the form.

where PC = (pij )i,j∈C, Q = (pij )i∈T,j∈C, R = (pij )i,j∈T . Here T the set of transient states and by C the set of recurrent states such that S = C ∪ T. Compute the matrix U = (I1 − R)
−1Q of absorption probabilities into the set of recurrent states.