Let X be a random variable that follows the exponential distribution with a rate parameter 1. (a) Compute P(X > 0.5|X > 0.3). (b) Show that the moment generating function of X is Mx(t) = iid Exponential(1). Use moment generating func- (c) Suppose that you have X1,., X30 tions to show that X; follows the Gamma distribution with shape parameter 30 and rate parameter 1. Hint: You may use the following fact without proving it: the mgf of Gamma-distribution with parameter a (shape) and 3 (rate) is (1- )-a. 25

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