Consider two independent random variables $X$ and $Y$, and $X \geq 0$, $Y \geq 0$. Let $Z = f(X, Y)$
Is it possible to calculate the partial derivative of $\mathbb{E}(Z)$ respect to $\mathbb{E}(X)$, i.e. $\frac{\partial }{\partial \mathbb{E}(X)} \mathbb{E}(Z)$?
More specifically, I would like to find what property $f(\cdot)$ should have to enforce the partial derivative is greater than 0, \begin{equation} \frac{\partial }{\partial \mathbb{E}(X)} \mathbb{E}(Z) \geq 0 \end{equation}
For example, polynomials with all positive coefficient should have it hold.
(Should I say $\{X\}$ is a (infinite) set of arbitrary R.V.s? Because the goal I'd like to achieve is to find some property of $f(\cdot)$ that $\mathbb{E}(f(X_1, Y)) \geq \mathbb{E}(f(X_2, Y))$ for any $X_1, X_2$ that $\mathbb{E}(X_1) \geq \mathbb{E}(X_2)$ )
I'd really appreciate for any hint.