I would like to know the $f(x)$ which minimizes the $\displaystyle\int_{-\infty}^{\infty} f(x)g(x)\,dx$.
Actually, this question start from the MMSE (Minimize Mean square error)
$$E[(X-g(Y))^2]=\int_{-\infty}^{\infty} E[(X-g(Y)|Y=y]^2 f_Y(y)dy $$
here, find the $g(Y)$ minimize the mean squre error.
and we can view as $ f(x) = E[(X-g(Y)|Y=y]^2 $ and $g(x) = f_Y(y)$.
So this questions is that how to find the $f(x)$ minimize the integral fomula.
I have no idea how to do it :)
Thank you for your answer in advance !
Assuming $g$ is not identically $0$, there is no such $f$: $\int_{-\infty}^\infty f(x) g(x)\; dx$ can be made arbitrarily large and negative.