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a) Demonstrate how the moments of a random variable x may be obtained from its moment generating function by showing that the rth derivative of E(e*) with respect to t gives the value of E(x") at the point where t = 0. Show that the moment generating function of the Poisson p. d. f. f(x) = = x! -; x E {0, 1, 2,...} is given by M(x, t) = exp{-p] exp{ue }, and thence find the mean and the variance. b) Demonstrate how the moments of a random variable may be obtained from the derivatives in respect of t of the function M(t) = Etexp(xt)}. If x = 1, 2, 3,... has the geometric distribution f(x) = pq* , wher q = 1 - p, . low that the moment pet generating function is M (t) = 1 -get . Hence find the mean and the variance of x