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Question (1 a)

An actuary wants to investigate if there is any correlation between students' scores in the CS1 mock exam and the CS2 mock exam. Data values from 22 students were collected and the results are: Student 1 2 3 4 5 6 7 8 9 10 11 CS1 mock score 51 43 39 80 56 57 26 68 54 75 72 CS2 mock score 52 42 58 56 47 72 16 63 48 80 68 Student 12 13 14 15 16 17 18 19 20 21 22 CS1 mock score 85 48 27 63 76 64 55 78 82 52 60 CS2 mock score 82 54 38 57 71 50 45 60 59 49 61 You are given that Ed0 for the mock score data using the Spearman's rank correlation coefficient and the Kendall's rank correlation coefficient along with normal approximations.1 Given that npr =0.3, my =0.4, mp= =0.6, find the probability that, of the lives (x), (y) and (2) (a) none will survive n years (b) exactly one will survive n years (c) at least one will survive n years. 2 Prove that (IA)ry = dry - d.(la)ry 3 Express in terms of Pr, Py; and pz the probabilities that, of three lives (x), (y) and (2), (a) all three will survive one year (b) at least one will survive one year (c) exactly two will survive one year (d) at least two will survive one year 4 Derive the formula ery = E(K) = > Pry where K = integer part of T (T = min{T1, T2}) t= 1 using Pr {K = k} = klqry. Evaluate A75:75 on the basis of A1967 - 70 ultimate at 4% interest. The probability that at least one of three lives aged 60 will survive to age 65 is eight times the probability that exactly one will survive to age 65. Assuming that the 3 lives are independent and subject to the same table of mortality, find the probability that exactly one life will survive to age 65. 7 (i) Define +Pry and show that Pry = tPr + tPy - tPry (ii) Hence, or otherwise, show that dry = ar tay - ary 8 12 years ago a man then aged 48 effected a without profits whole life assurance for f10,000 (payable at the end of the year of death) by annual premiums. The premium now due is unpaid. He now wishes to alter the policy so that the same sum assured will be payable at the end of the year of the first death of himself and his wife, who is 4 years older than himself. Calculate the revised office annual premium, ceasing on the first death, if the office uses the following basis for premiums and reserves. mortality: A1967-70 ultimate, rated down 4 years for female lives, interest: 4% per annum, expenses: 3% of all office premiums including the first, with additional initial expenses of 15% of the sum assured. (This additional initial expense is not charged again on the conversion of an existing policy, providing that the sum assured does not increase.) 9 Consider the random variable L equal to the present value of f1 payable immediately on (i) the first death of (x) and (y), and (ii) the second death, in each case at a given rate of interest i p.a. Show that, in case (i). var(L) = Ary - (Any)? where * indicates a rate of interest of 2i + 1 p.a., and give a corresponding result for case (ii)