Provide step by step solution to the following questions. Please be precise.

Prove each of the following results: (i) The probability that exactly x decrements will occur in a population consisting initially of n individuals subject to a single decrement with rate q per annum is: [3] (ii) The maximum likelihood estimator of the parameter q for the binomial model equals the number of decrements divided by the initial population. [4] iii) The maximum likelihood estimator of the parameter / for the Poisson model is an unbiased estimator of the force of decrement. [3] [Total 10](i) Define the accumulation factor A(,/+h) and give a formula for the force of interest 6() per unit time in terms of the accumulation factor. [2] (ii) The force of interest 6() at time r (measured in years) is given by 6(1) =0.01/ +0.04. (a) Calculate the corresponding nominal rate of interest for the period / =1 to 1 =2. (b) If an investment of 1 is made at time / = - , calculate the value to which it will have accumulated by time r =6. [6] (iii) Calculate the accumulated value after 6 months of an investment of $100 at time 0 at the following rates of interest: (a) a force of interest of 0.05 pa (b) a rate of interest of 5% pa convertible monthly (c) an effective rate of interest of 5% pa. [3] [Total 1 1]