Solve the following:

Invertability of the Laplace-transformation. Let f: 0, co) > R be a continuous and bounded map. We define the Laplace-transform f of f as f ( 1 ) : = / ex f (y )dy . (a) Let X1, X2, ... be i.i.d. random variables with distribution EXP(A), i.e. P(Xk > k) = e-Ax; x > 0, E(Xk) = 1-1, and D'( Xk) = 1-2. Show that E(f (Sn)) = (-1)2-1Anf (n-1)(x) (n - 1)! where Sn = X1+ . .. + Xn and f("-1) denotes the n - 1th derivative of f. (b) Prove that the Laplace transform satisfies the following inversion formula (for all continuous and bounded map f ): f(y) = lim (-1)2-1(n/y)"f(2-1)(n/y) (n - 1)