Refer to the attachments below to answer the questions that follow. Kindly explain your responses.

In a process, the number of failures before the first success has the following probability distribution: P(X = x) = 0(1-0)* x = 0,1,2. ... Show that the method of moments and the method of maximum likelihood both lead to the same estimator for 0 . [5]A special endowment policy pays a sum assured of $20,000 to a life who is currently aged exactly 57 after three years or at the end of the year of earlier death. Annual reversionary bonuses are declared at the end of each policy year, and a terminal bonus is payable at maturity only. Policies may be surrendered only at the end of each policy year. On surrender, the policyholder receives a return of premiums with interest calculated at the rate of 3% per annum. A level premium of 18,000 is paid at the start of each year. The premium basis is as follows: Interest: 7% per annum . Mortality: AM92 Select Surrender rates: 15% of all policies in force at the end of year 1 5% of all policies in force at the end of year 2 . Reversionary bonuses: 6% per annum compound Terminal bonus: 10% of all other benefits payable at maturity . Expenses: Initial $500 Renewal (30 at start of year 2 E35 at start of year 3 Termination $100 per termination (death, surrender or maturity) Reserves: Net premium reserves, using AM92 Ultimate mortality and 4% per annum interest. (i) Calculate the profit signature for this policy according to the premium basis. [14] (ii) Explain briefly whether you think the company expects to declare the bonus rates it has assumed in its premium basis, assuming all the other assumptions in the basis are realistic. [2] [Total 16]