Consider a random variable X with quantile function Q_X(p).

(a) Let Y = g(X), with g a strictly increasing and continuous function. Show that Q_Y(p) = g(Q_X(p)). (i.e., the quantile function of a strictly increasing transformation is the transformation of the quantile function.)

(b) Now let Y = g(X), with g a strictly decreasing and continuous function. Further suppose that the CDF F_X of X is strictly increasing on its support X. How does Q_Y(p) relate to Q_X(p) in this case?

(c) Suppose that Q_X(p) is differentiable at every point p (0, 1). Using the formula for the derivative of an inverse function, show that Q'_X(p) = 1/(f_X(Q_X(p))) . [Note: Q_X can be differentiable only if F_X is strictly monotone and differentiable on the support X of X.]

Note: A quantile can be thought of as the inverse CDF.