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Define the force of mortality. ,, of a random variable 7 denoting length of life. [1] The mortality of a certain species of animal has been studied. It is known that at ages under five years the force of mortality, , is constant. (ii) Write down an expression, in terms of u, for the probability that an animal will survive from birth to exact age five years. [1] Mortality of these animals at ages over five years exact is incompletely understood. However it is known that the probability that an animal aged exactly five years will survive until exact age 10 years is twice the probability that an animal aged exactly five years will survive until exact age 20 years. Assume that the force of mortality, A, is constant at ages over five years exact. (iii) Calculate 2. (3] (iv) Calculate the expectation of life at birth for these animals if 2 = p. [11 (v) Derive an expression. in terms only of u. for the expectation of life at birth for these animals if 2 # p. [4] [Total 10]The cumulative incurred claims for an insurance company for the last four accident years are given in the following table: Development year Accident year 2 3 2005 06 136 140 168 2006 100 156 160 2007 120 130 2008 136 It can be assumed that claims are fully run off after three years. The premiums received for each year from 2005 to 2008 are 175, 181, 190 and 196 respectively. Calculate the reserve at the end of year 2008 using: (a) The basic chain ladder method. (b) The Bornhuetter-Ferguson method. [12]