Need this as a part of a bigger thing I'm trying to prove, but couldn't have it still, after a long time trying.
$A,B,C$ are three independent random variables. We define additional two random variables: $f(a) = P(a > B), g(a) = P(a > C)$. Prove that $COV[f,g]=0$.
Note that $f,g$ are not necessarily independent...
Suppose you are trying to prove: given r.v. $A$, construct two r.v.s $A'=F_{B}(A)$ and $A''=F_{C}(A)$, where $F_{B}$ and $F_{C}$ happen to be cdfs of two r.v.s $B$ and $C$, then $A'$ and $A''$ have zero covariance.
I do not think this is correct. Take $A$, $B$, $C$ uniform on $[0,1]$. Then cdf is $F_{B}(x)=F_{C}(x)=x$ on $[0,1]$ and hence $A=A'=A''$.