1. Consider the distribution function F(X):0 for x < ? 11/10 for -1 ? x < 2 x^3/48 for for 2?x<39(x-1)^2/64 for 3?x<11/3 1 for x ? 11 / 3a) Assuming that a uniform random number generator is available, use the inversion method to provide an algorithm for generating a random variable X having distribution function F .—I think I understand this part, but am really struggling with the rest of the question—b) Compute E[ X ] and Var[ X ] analytically.c) Prove the following converse of the inversion method: if X is a random variable having acontinuous, strictly increasing distribution function F, then U = F ( X ) is uniformly distributed on [0,1]. [Hint: the assumptions on F ensure that F?1 (F(x))= x for any real number x, andF(F?1(y))= y for any y?[0,1]. Also note that both F and F?1 are nondecreasing functions.]
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