I have a normally distributed vector $(X,Y)$
$$(X,Y) \sim N\biggr[\begin{pmatrix} 1 \\ 2\end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} \biggr]$$
TODO: Find such measurable function $g(x)$ that minimize expectation $$E[-24Y^{4} + 2g(X)Y^{2} + 2(g(X))^{2}+4] \longrightarrow min$$
My attempt:
It's straightforward to notice that
$$E[-24Y^{4} + 2g(X)Y^{2} + 2(g(X))^{2}+4] = E[(Y^{4} + 2g(X)Y^{2} + 2(g(X))^{2})-25Y^{4}+4] = \\ = E[(Y^{2} + g(X))^2 + (4-25Y^{4})] = E[(Y^{2}+g(X))^{2}] + E[4-25Y^{4}]$$
Here we may consider only first term because second is a constant. So,
$$E[(Y^{2}+g(X))^{2}] = \int \limits_{-\infty}^{+\infty} t^{2}f_{[Y^{2}+g(X)]}(t)dt$$
Question:
And here I stumbled because I don't know how to detach random variables from random vector and find density function. I think that's technical task, but I don't know how to properly solve it.
PS: If you've found some mistakes in my steps - let me know, please.