Answer the following question in the attachment below.

Consider the following time-inhomogeneous Markov jump process with transition rates as shown below: 2 0.1 0.21 0.1/ 0.21 4 0.05r 0.5 (i) Write down the generator matrix at time /. [2] (ii) Write down the Kolmogorov backward differential equations for P;(s,r) and Pi3(s,1). [3] (iii) Using the technique of separation of variables, or otherwise, show that the solution of the differential equation for By(s,() is: 12 0.25(12-5-) B3(s, !) =e [4] (iv) Show that the probability that the process visits neither state 2 nor state 4 by time /, given that it starts in state 1 at time 0, is: 8 -0.07512 _1 -0.251- [6] (v) State the limiting value as / -> of the probability in (iv). Explain why this must be the case for this particular model. [2] [Total 17]