Solve the three questions below. All the information is provided.

Calculate the present value as at 1 January 2005 of a 5-year annuity consisting of payments of $100, starting on 1 January 2005 and payable on the first day of each month with an "R" in its name. Assume a constant annual effective rate of interest of 8%.{i} {ii} If the moment generating function of X is M; [t] , then derive an expression for the moment generating function of 21' + 3. [2] Hence, if I is normally distributed with mean ,u and variance 62' derive the distribution of 2.3:\" +3. [2] [Total 4] (i) I, denotes the future lifetime of a life currently aged x. Write down the probability density function of Ty - [1] (ii) Using your answer to (i), show that: a (a) as log Px = -Ax+s; and (b) [4] (iii) In a certain population, the force of mortality is given by: Hx 60 80 0.025 Calculate the probability that a life aged exactly 65 will die between exact ages 80 and 83. [3] [Total 8]