Answer the following questions.

a - nv (i) Show that (10)= = [4] (ii) A 25-year annuity certain provides payments annually in arrear of $20 in year 1, f30 in year 2, f40 in year 3, and so on, until the payments reach $100. Payments then decrease by f5 each year until the end of the term. Calculate the present value of the annuity at 6%% per annum effective. [4] [Total 8]The random variable Y' has a gamma distribution with parameters o (> 1) and 1. (i) (a) Show that the mode of Y is given by: (b) By considering the relative locations of the mean and mode using sketches of the gamma distribution, state how you would expect the distribution to behave in the limit as o -> , but where 1 is varied so that the mean has a constant value . [5] (ii) Given that or =50 and 1 = 0.2, calculate the value of P() > 350) using: (a) the gamma-chi square relationship (b) the Central Limit Theorem. [5] (iii) Explain the reason for the difference between the answers obtained in part (ii) [1] [Total 11]