(a) Let X follow an exponential distribution with parameter A = 1. Denote the mean value and variance of X by ,u = 1 and a2 = 1 respectively. Compute the approximate expectation of X3 using the 2nd order moment approximation ]E[q$(X)] 1-: put) + q\"(u)02 with ($) = 3:3. Sketch the graph of 925 as well as the approximating function obtained from 2nd order Taylor approximation about M. Le. about a: = 1. [4] Compute ]E[X3] exactly. e.g. by integration or by using the mgf. [TYPE:] Discuss the direction of the deviation of the approximation computed above from the true value. [3] (b) Let W and Z be independent both following a U(0, 7r/2) distribution. Compute IE[sin(W + 2)] using the 2nd order moment approximation. Hint: write V = W + Z. What are ]E[V] and Var(V)? [3] For functions q5(-) and 1M), write down the definition of ]E[(W)(Z)] as an integral. By considering the form of this integral, explain carefully why El(W)(Z)l = ]E[45(W)]1E[TJJ(Z)]- [2] Compute ]E[sin(W + 2)] exactly. How accurate is the 2ndorder moment approximation from part (b)i compared to the one in part (a)? Hint: Re membering trigonometric identities will help you use the result from part (b)ii here. [3]