Let t0 > 0 and X be a random variable such that E(etX) < for all t R with |t| < t0. Let (t) = log(E(etX)) for |t| < t0. From one of our theorems in class, we know that has derivatives of all orders, so you may use this in your solutions.

(a) What is (0)?

(b) Write (0) in terms of the moments of X.

(c) Write (0) in terms of the moments of X.

(d) Write (0) in terms of the moments of X.

(e) Write out the first four terms (i.e. up terms of degree 3) in the Taylor series for (based at 0).