Kindly it's a microeconomics questions

1.1 Given that mp, =03, spy =0.4, ap, = 0.6, find the probability that, of the lives (r), (y) and ia) none will survive n years (b) exactly one will survive n years ic) at least one wall survive n years. 1.2 Prove that (IA)n = dry - d.(la); 1.3 Express in terms of pr, po, and p. the probabilities that, of three lives (r), (y) and (=), (a) all three will survive one year b) at last one will survive one year ic) exactly two will survive one year id) at least two will survive one year 1.4 Derive the formula er = E(K)= ) .my where K = integer part of T (T = min(Ti, 12)) 2= 1 using Pr(K = ] = =|417 1.5 Evaluate A,? on the basis of A1967 - 70 ultimate at 4% interest. 1.6 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 1.7 (i) Define up,, and show that they = cp, + ty - Pry ni) Hence, or otherwise, show that an = a, + dy - dry 8 12 years ago a man them aged 48 effected a without profits whole life assurance for 610,000 ipayable 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: A 1967-70 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.) 1.9 Consider the random variable & equal to the present value of fl payable immediately on (i) the first death of (=) and (), and (ii) the second death, m each case at a given rate of interest i p.a. Show that, in case (i). vari [) = Ar- (An)' where * indicates a rate of interest of 2/ + 7 p.a., and give a corresponding result for case (ii)