Density and Distribution of Fx(X)

Question

This problem's been driving me up the walls. I'm not sure where to go with it. Let X be a continuous random variable with density $f_X(x)>0$ for all x. Let $F_X$ be the distribution function for X.

a) Find the density and distribution functions for $U=F_X(X)$.

b) Find the density and distribution functions for $V=log(F_X(X))$.

Answer 1

We give a brief solution of the first problem.   The second yields to similar tools.   After we know the answer to the first problem, we no longer need to work with the potentially confusing $F_X(X)$.

Let $U=F_X(X)$. Then for $0\lt u\lt 1$, we have $$\begin{align}F_U(u) & =\Pr(U\le u) \\ & =\Pr(F_X(X)\le u) \\ & =\Pr(X\le F_X^{-1}(u)) \\ & =F_X(F_X^{-1}(u)) \\ & = u\end{align}$$

Thus $U$ has uniform distribution on $(0,1)$.

Now find $V= \log U$ ...