Let $X$, $Y$, and $Z$ be continuous random variables. $X$ and $Y$ are independent with respect to each other, while $Z = g(X, Y)$.

$E(X)$, $E(Y)$, and $E(Z)$ are known. Marginal pdf's $f_X(x)$, $f_Y(y)$, and $f_Z(z)$ are known. Joint pdf's not given.

I am attempting to compute the covariance $Cov(Z, X)$. I know that $$ Cov(Z, X) = E(ZX) - E(Z)E(X) $$ but I am unsure how to compute $E(ZX)$ with $Z$ dependent on $X\ \ $ ($Z = g(X, Y)$).

What is the approach to evaluating the covariance of two dependent random variables?

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    $\begingroup$ It is hard to believe that any non-trivial estimates can be made unless we know more about $g$. For example, for $Z = aX + bY$ with appropriate $a$ and $b$, the correlation can have any value between $-1$ and $1$. $\endgroup$ – Sangchul Lee Oct 7 '16 at 1:27

Clearly if $Z=g(X,Y)$, the marginal probability density functions of $X,Y$ are known ($f_X, f_Y$), and these two random variables are independent, then:

$$\mathsf E(XZ) = \int_\Bbb R\int_\Bbb R x\, g(x,y)\,f_X(x)\,f_Y(y)~\operatorname d y\operatorname d x$$

But of course that requires you to know $g$, and I can't see any way to escape that requirement.


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