It is well known that for two increasing functions $f$,$g$ and for any random variable $X$ then $cov(f(X),g(X))\geq{}0$. Now assume $f,g$ have the same domain $D$ and let $A\subset{}D$. What can I say about the sign of $cov[f(X),g(X)/A]$.
1 Answer
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If the set $A$ has positive probability, than conditional covariance works essentially as unconditioned one because all integrations involved are performed with respect to $P(.|A)$ which is a well-defined probability measure. ($P$ is the original probability measure)
