Let X be a random variable in the range (0,1), i.e. P(X)=0 for X<=0 or X>=1. Is there a way to prove that Cov(X,X/(1-X)) > 0 ? Is it true for any pdf f(X) whose support is in (0,1)?
Wow, this took me a surprisingly long time to show. Yes, it is true, provided that you allow the inequality to be attained (if X is a point mass you get a covariance of 0).
Let . Let and . First, note that is convex, and hence by Jensen's Inequality we have .
Next, note that it suffices to show . By definition this is the statement that
and solving for this is the statement , which is always true by Jensen's Inequality.