Please answer this question with all steps. Thanks!

A few unrelated questions. Justify each of your answers, this means prove or give a counterexample for each of the questions. a) Let X be a continuous random variable with distribution Fx. Does there exist a random Y such that its distribution Fy satisfies FY(x) = 2Fx(x)? b) Let X ~ N(0, 1) and Y ~ N(0, 1) be independent. Then X2 + Yz is an exponential random variable. c) Let X and Y be two jointly continuous random variables with joint distribution Fx,y and marginal distributions Fx and FY, respectively. Suppose that Fx,y (a, b) = Fx(a) FY(b) for every a, b E Z. Does this imply that X and Y are independent