Let $Z$ be a standard Normal random variable. Let $F$ be a continuous random variable such that
$$g_{F|Z}(f|z)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{(f+2z)^2}{2}\right).$$
What is $\mathbb E[F|Z=z]$ (as a function of $z$)?
I tried to find it by treating $z$ as a constant and used the Normal mean $\mathbb E[F \mid z] = -2\cdot z$.
And then what is $\mathbb E[F]$? What is $\mathrm{COV}(F,Z)$? I can’t find the variance.
The expectation of the product is $\mathbb E[FZ] = \mathbb E[Z\cdot \mathbb E[F \mid Z]]$
But I can’t find the values.
The conditional PDF of $Z$ given $F=f$ is of the form
$$\alpha(f) e^{−\mathrm{quadratic}(z,f)}$$
By examining the coefficients of the quadratic function in the exponent, can one find $\mathbb E[Z \mid F=f]$ and $\mathbb{V}(Z \mid F=f)$?