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1. (a) Let X follow an exponential distribution with parameter ) = 1. Denote the mean value and variance of X by μ= 1 and σ^2=1 respectively. i. Compute the approximate expectation of X^3using the 2nd order moment approximation E[ø(X)] ≈ φ(µ) + }φ"(μ)σ² with (x) = x³. Sketch the graph of φ as well as the approximating function obtained from 2nd order Taylor approximation about μ,i.e. about x=1. Compute E[X^3] exactly, e.g. by integration or by using the mgf. Discuss the direction of the deviation of the approximation computed above from the true value.
1. (a) Let X follow an exponential distribution with parameter 1 =1. Denote the mean value and variance of X by u = 1 and o2 = 1 respectively. i. Compute the approximate expectation of X3 using the 2nd order moment approximation Elo(X)] − o(u) + 10"()o2 with $(x) = x3. Sketch the graph of o as well as the approximating function obtained from 2nd order Taylor approximation about h, i.e. about x = 1. [4] ii. 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]