1、Suppose       Y     aT   is the random present value variable for a continuous life annuity of $1 per
     year   to   (x).   Assume δ 0.05   and   that   the   future   life   time   variable   T   has   p.d.f   given   by
     f T (t)   0.015e−0.015t ,t =≥0
      a)  calculate   ax by either the aggregate or payment method
      b)  What is the probability that a fund of       ax  at age x is sufficient to withdraw $1 per year
          continuously for us as long as (x) lives?
      c)  What is the possible range of value of Y?
      d)  What event in terms of T is equivalent to 5 ≤Y ≤10 ?
      e)  What is the    Pr(5 ≤Y ≤10) ?
                               2
2  、If       Ax    0.06 and      Ax     0.01 and Yis the present value variable for a             continuous
$1000 per year life annuity to (x), find     E (Y ) and δ assuming δ =0.05 .
                                                           T
3、Suppose 100 lives age x are paid $1000 per year continuously while the survive. How big a
fund   is   needed   at   the   time   of   simultaneous   issue   in   order   to   by   90%   certain   of   having   enough
money to pay the life annuities? Assume that the present value variable Y is as in Qusetion 2 and
use the central limit theorem on Yagg, the aggregate present value variable.