An insurance contract based on the failure time t pays 1

Question

1. A continuous failure time has a density function, f_T (t) = β²te-^βt (a gamma distribution with first parameter 2). The force of interest is a constant δ. Find expressions in terms of β and δ for: (a) A_T‾ ; (b) a_T‾ ; (c) the net annual rate of premium payment when premiums are payable continuously prior to failure, for an insurance paying 1 at failure and (d) the reserve at time k for the contract in (c).

2. An insurance contract, based on the failure time T, pays 1 unit at the moment of failure provided this occurs within 5 years. Nothing is paid for failure after that time. The force of interest is a constant 0.1. If T has the hazard function

μ_T (t) =2/10 - t,

Find the probability that the present value of the benefits is strictly positive, but less than or equal to e^-0.3.