Answer the following questions clearly explaining your answers. All the information is provided.

The random variable S has a compound distribution so that S = Xj+ ...+ Xy (and S=0 if N =0) where the X,'s are independent, identically distributed (as a variable X ) and are also independent of N. (i) Use the result var[Y] = E[var(Y | X )] + var[E(Y | X )] to obtain an expression for the variance of S in terms of the means and variances of the random variables X and N . [4] (ii) / has a binomial distribution with parameters n =100 and p =0.05 and X has an exponential distribution with mean 50. Calculate the mean and standard deviation of S. [2] (iii) By using a suitable approximation, calculate the probability that S exceeds 500. [3] [Total 9]Ken and Barbie, aged 60 and 64 respectively, take out a policy under which the benefits are: A lump sum of $50,000 payable at the end of the year of the first death provided this occurs within 10 years. An annuity payable annually in advance with the first payment due to be made 10 years from the date of issue. The annuity will be of $10,000 pa for so long as both Ken and Barbie are still alive or $5,000 while only one of them is alive. Level premiums are payable annually in advance for at most 10 years and will cease on the first death if this occurs earlier. Calculate the amount of the annual premium on the following basis: Interest: 4% pa Mortality: Ken: PMA92C20 Barbie: PFA92C20 Expenses: None [8]