Read through the attachments below and answer all the questions that follow clearly giving explanations where necessary.

Consider the following time-inhomogeneous Markov jump process with transition rates as shown below: 2 0.1 0.21 0.Ir 0.21 4 0.051 0.50 (i) Write down the generator matrix at time /. [2] (ii) Write down the Kolmogorov backward differential equations for By(s,!) 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: Pig(s, 1) =e -0.25(13-$2) [4] (iv) Show that the probability that the process visits neither state 2 nor state 4 by time /, given that it starts in state I at time 0, is: 8-0.07512 1 -0.2513 [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]Discuss how time selection and class selection can affect the results of a mortality investigation and how you consider these factors should be allowed for in developing mortality tables for practical use. [6]Let i, be the effective interest rate in year /. Find the expected value and standard deviation of the accumulated value at time 3 of investments of 1 at time 1, 2 at time 2 and 3 at time 3, given the following information: E[i ] = 4% standard deviation of i = 0.5% Eliz ]= 4.5% standard deviation of i, = 0.8% E[i; ] = 4.8% standard deviation of is = 1% Interest rates in different years are independent. [8]