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3. (a) Prove that for a random variable with expectation u and variance o2, 2 lim t->0 log M(t) - t = 2/2 t2 2 (b) For a random variable X that has a chi-square distribution with 4 degrees of freedom, the moment generating function is 1 M(t) = = (1 - 2t) 2 . Use this moment generating function to show the expectation E(X) = 4. (c) Calculate the variance of X. (Thanks to part(a), there are two alternative ways to do this).

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