1. A failure time T is uniformly distributed on the interval [0, 20]. The force of interest is a constant 0.05.

(a) A deferred insurance contract provides for a payment of 1 at the moment of failure provided that this occurs after time 5. If Z‾ is the present value of the benefits, find the 80^th percentile of Z‾. That is, find the point z such that P(Z‾ ≤ z) = 0.80.

(b) Find the 80^th percentile of Z‾ when the contract is a term insurance policy that pays 1 unit if failure occurs in the first 5 years.

2. Consider a deferred term insurance contract. It provides for 1 unit payable at the moment of failure provided that this occurs between N and 2N years from now. Nothing is paid if failure occurs in the first N years or after 2N years. You are given that the force of failure μ and the force of interest δ are constants such that μ = δ/2. Moreover, you are given that e^-Nδ = 0.36. If Z is the present value of the benefits, calculate FZ (z) for all nonnegative values of z.