Hi tutors. Help me out.

The distribution of the number of claims, N, from a risk is zero-truncated geometric, with probability function P, = ap,-1 for n = 2, 3, 4, ..., and individual claim amounts, X, have probability function {fx),. Let (gx ), denote the probability function of aggregate claims, S. (a) Show that P (r) = 1 - a tarP (r) +aPN(r). (b) Starting from Ps(r) = PN [Px(r)], show that PS(r) = (1 -a) P (r) +a (Px(r)P; (r) + Ps(r) PX(r)). (c) Find a recursion formula for gx, x = 1, 2, 3, ..., and show that 80 = (1 - @ ) fo/ (1 - a fo). (d) Use your answer to (c) to show that E [S' ] = E [X' ] + , OF[S']E[X-] The probability function (P,), of the number of claims, N, from a risk satisfies the recursion formula bi Pn = Pn-i (4.37) n where p, = 0 for n [air'PM()+ (ia; + b; )r- PN(r)]. i=1 (b) Let Y, = _ _ X; for i = 1, 2, ... , k. Find an expression for Pi (r) in terms of Px (r) and Px (r). (c) Show that k b; P's(r) = a; Pr(r) P;(r) + i+ Pi (r) Ps(r). 1= (d) Show that for x = 1, 2, 3, ... k E bij 8x-j fi* (4.38) ix =1 and that the starting value for this recursion formula is go = PN(fo). (e) What is the main problem associated with the use of formula (4.38)