(30 points) Fix x 2 0, and define an " Esscher-transformed" frailty random variable As with probability density function (or discrete probability mass function in the discrete case) fAs (A) = e-safA(X)/MA(-s), 120. (i) Show that A has moment generating function. MA, (2) = E(ezAs ) = MA(z - 3) MA(-S) (1) (ii) Define the cumulant generating function of A to be CA(2) = In[MA(=)], (2) and use (i) to prove that CA(-s) = E(As) and c'X(-s) = Var(As). (3) (iii) For the frailty model with survival function, prove that the associated hazard rate may be expressed as hx(x) = a(x)CA(-A(x)), where cA is defined in (ii). (iv) Use (iii) to show that h'x(x) = a'(x)CA(-A(x)) - [a(x)]2ck(-A(x)). (4) (v) Prove using (iv) that if the conditional hazard rate hxjA(x|A) is nonincreasing in a, then hx(x) is also nonincreasing in r