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Math Help - Covariance of functions of a random variable

  1. #1
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    Covariance of functions of a random variable

    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)?
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  2. #2
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    Re: Covariance of functions of a random variable

    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 f(x) = \frac x {1 - x}. Let \mu = \mbox{E}[X] and \tau = \mbox{E}[f(X)]. First, note that f(x) is convex, and hence by Jensen's Inequality we have \tau \ge f(\mu).

    Next, note that it suffices to show \mbox{Cov}[1 - X, f(X)] \le 0. By definition this is the statement that

    \displaystyle \mbox{E}[(1 - X)f(X)] - \mbox{E}[1 - X]\mbox{E}[f(X)] = \mu - (1 - \mu)\tau \le 0

    and solving for \tau this is the statement f(\mu) \le \tau, which is always true by Jensen's Inequality.
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