Question: 3. Let Mx (t) be the moment-generating function of a random variable X and be the mean of X. (a) Show that the moment-generating

3. Let Mx (t) be the moment-generating function of a random variable X and be the mean of X. (a) Show that the moment-generating function of (X ) is Mx- (t) = et Mx (t). (b) Show that the rth derivative of Mx- (t) with respect to t at t the rth moment about the mean of X. = 0 gives (c) Given that the moment-generating function of a normal random variable X with mean and variance is Mx (t) = exp (t+2) +10). Using (a) and (b), find the skewness and the kurtosis of X, which are respectively, E ((X - )) E ((X )) - 3 and 4 = 03 04

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