Expected Value of Two Random Variables

Question

X is a random variable with a probability density function $f(x)$, g(x,y) is a function of two variables one of them is the random variable. I have \begin{equation} \int_{-\infty}^{\infty} g(x,y)f(x)dx = c, \forall y \in \mathbb{R}. \end{equation} Can we imply anything about the relation between g(x,y) and y?

Answer 1

I'm not sure about what you mean about "the relation between $g(x,y)$ and $y$", but I am going to answer what I understand of it: "can we infer than $g$ does not depend on $y$?"

The answer is no: take $f(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$ (pdf of a standard Gaussian), and let $$ g(x,y) \stackrel{\rm def}{=} e^{xy-\frac{y^2}{2}}. $$ Then $\int_{\mathbb{R}} g(x,y)f(x)dx = 1$ for all $y\in\mathbb{R}$, yet $g$ does depend on $y$ in a non-trivial fashion.

(It is also easy to build examples where $g$ does not depend on $y$, of course, for the other statement "$g$ does not necessarily depend on $y$ either.")