3. Change of Variables Let X be a continuous random variable with cdf FX and pdf fX > 0 everywhere, and let Y = g(X), where g is a differentiable function. a. Suppose that g is also invertible. Find the pdf of Y , fY , in terms of g and fX . (Hint: Does the invertibility of g imply that it has another property? Your answer should be broken up into two cases.) b. Let U Uniform([0,1]). Using the conclusion from part a, show that F1(U) has the X same distribution as X. (This allows us to generate a given random variable given only a uniform random number generator.) c. Now suppose that g(x) = x2. Find the pdf of Y in terms of the pdf of X. Also find the pdf of Y when X is a standard normal random variable in particular. (Note that this g is not invertible, unlike in part a.)