Question

1. Consider a three-state continuous-time Markov chain in which the transition rates are given by Q = egin{pmatrix} 0 & 2lambda & 0 \ lambda & 0 & lambda \ 0 & 2lambda & 0 end{pmatrix} The states are labelled 1, 2 and 3. Throughout this problem and whenever confronted with an ordinary differential equation of the form x'(t) = ax(t)+b(t), it might be beneficial to consider the function y(t) = e^{-at}x(t). (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Write down all nine forward equations. (c) Write down the backward equations for p12 and p21. and use the symmetry of Q to solve these equations. (d) Deduce the remaining pij(t). (e) Obtain the steady-state probabilities of this Markov chain in two different ways.

          1. Consider a three-state continuous-time Markov chain in which the transition rates are given by
Q = egin{pmatrix} 0 & 2lambda & 0 \ lambda & 0 & lambda \ 0 & 2lambda & 0 end{pmatrix}
The states are labelled 1, 2 and 3. Throughout this problem and whenever confronted with an ordinary differential equation of the form x'(t) = ax(t)+b(t), it might be beneficial to consider the function y(t) = e^{-at}x(t).

(a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states.
(b) Write down all nine forward equations.
(c) Write down the backward equations for p12 and p21. and use the symmetry of Q to solve these equations.
(d) Deduce the remaining pij(t).
(e) Obtain the steady-state probabilities of this Markov chain in two different ways.

1. Consider a three-state continuous-time Markov chain in which the transition rates are given by
Q = eginpmatrix 0 2lambda 0 lambda 0 lambda 0 2lambda 0 endpmatrix
The states are labelled 1, 2 and 3. Throughout this problem and whenever confronted with an ordinary differential equation of the form x'(t) = ax(t)+b(t), it might be beneficial to consider the function y(t) = e^-atx(t).

(a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states.
(b) Write down all nine forward equations.
(c) Write down the backward equations for p12 and p21. and use the symmetry of Q to solve these equations.
(d) Deduce the remaining pij(t).
(e) Obtain the steady-state probabilities of this Markov chain in two different ways.

Added by Jeremy R.

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