Give clear explanations.

6. Let F(x) = 1 -e-*x, x 2 0. Use equation (8.24) to show that a - Rs RS = 2cRs - ca + + 8 Differentiate this to find an expression for Ry and hence show that -202 Ro = (ac - 2)3 Hence show that E [TJ 20'a + 2cx(2ca - A)u + 12(ca - 2 )u2 c2 (ca - 1)3 7. Lundberg's fundamental equation is At8- cs = 2f*(s). (8.39) (a) Show that when & > 0 there is a unique positive root of this equation, and that this root goes to 0 as & goes to 0. (b) Show that when f(x) = de "x, x > 0, then equation (8.39) is the same as equation (8.24). 8. Let *(s, 8) = e-s" p(u, 8) du. (a) By taking the Laplace transform of equation (8.21) show that up*(s, 8 ) = _12-2f*(s) - cso(0, 8) "s At8-cs-2f*(s) (b) Deduce that 4(0, 8) = y(0)k*(PS) where ps is the unique positive root of Lundberg's fundamental equation, and k is given by equation (8.1). (c) Show that m2 E [To.c) = 2m (c -Am1) 9. Consider the dividends problem of Section 8.7 and let the initial surplus be b, so that dividends are payable immediately and this dividend stream ceases at the time of the first claim. (a) What is the distribution of the amount of dividends in the first dividend stream?(b) Let / denote the number of dividend streams. Show that Pr(N = r) = p(b)-' (1 - p(b)) for r = 1, 2, 3, ... where p(b) = f(x)x(b - x, b)dx. (c) Find the moment generating function of the total amount of dividends payable until ruin, and hence deduce that the distribution of the total amount of dividends payable until ruin is exponential with mean C A(1 - p(b)) (d) Suppose instead that the initial surplus is u