Provide solution to the following questions. Include step by step explanations.

A mortality investigation covered the period 1 January 2007 to 1 January 2008. Time is measured in years from 1 January 2007 and P.() denotes the number of lives at time / aged x last birthday. The following data were recorded for each x : d, = number of deaths aged x next birthday P. (0) and P.(1) (i) Obtain an expression for the initial exposed to risk in terms of the available census data that may be used to estimate the initial rate of mortality qx+. stating your assumptions. (ii) Determine the value of , stating any assumptions you make.A group of six lives was observed over a period of time as part of a mortality investigation. Each of the lives was under observation at all ages from age 55 until they died or were censored. The table below shows the sex, age at exit and reason for exit from the investigation. Life Sex Age at exit Reason for exit M 56 death F 62 censored F 63 death DUAWN M 66 death M 67 censored M 67 censored The following model has been suggested for the force of mortality: #(x | Z = =) = /(x)eB: where: x denotes age Mo (x) is the baseline hazard 2 = 0 for males and z = 1 for females. Write down the partial likelihood for these observations using the model above.A lump sum of $14,000 will be invested at time ( for 4 years at a constant annual rate of interest i. (1+/) has a log-normal distribution with mean 1.05 and variance 0.007. What is the probability that the investment will accumulate to more than $20,000 in 4 years' time? [5]