Answer all questions below

Example 7.5.3. For several years an insurance company has issued a large number of special en- dowment assurances. Each policy matures at exact age 65 and is effected by annual premiums payable on each January 1st throughout the term. The sum assured, payable at the end of the year of death during the term, is one half of the sum assured that will be paid if the policyholder survives to age 65. Details of the policies in force on 31 December 1988 are as follows: Exact age Total sums assured Total annual premiums payable on death (f) (f) 55 3500000 250000 The claims in 1989 were on policies with the following total sums assured and premiums: Total sums assured Total annual payable on death (f) premiums (f) 50000 4000 Assuming that at 31.12.88 the office's funds were equal to its reserves, calculate the mortality profit or loss in 1989, given that the following reserving basis is used: prospective method, gross premium reserve using A1967-70 ult., 4% p.a. interest, no expenses.The random variables X1,...,X represent independent observations from a population with mean p! and variance 0'2. [i] State the mean and variance of the sample mean I? = i211. . [2] [ii] Show that gal. 1?): =21? nil. [2] (iii) Determine the expected value of gut} E): . [3] [iv] Hence showthat E[32]=a'2, where SZ=LXE Hf2]. [l] [v] For the special case where the parent population is 1101111311}? distributed and the values of the population parameters la and o2 are known, state two results that 3'! specify the precise distributions of f and i2: (Xi f )2 . [2] C" i=1 [Total 10] The random variable S has a compound distribution so that S = Xj+ ...+ Xx (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) N 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]