Given random variables $X \sim N(0, \sigma^2_x)$ and $Y \sim N(X, \sigma^2_y)$, solve $\mathbb{E}[Y|X]$.
My attempt:
$\mathbb{E}[Y|X] = \int_{-\infty}^{\infty} y \; P_{Y|X} (Y|X) \; dy$
But I do not have the expression for $P_{Y|X} (Y | X)$, just the normal distributions for the $x$ and $y$.
Is the expectation of a normally distributed variable conditioned on its mean equals to the expectation of the variable? That is:
$\mathbb{E}[Y|X] = \mathbb{E}[Y] = X$
I don't see how conditioning on the mean (not its value) would change the expectation of $Y$.
I also tried expressing $Y$ in terms of $X$ by substituting the normal distribution into $\mathbb{E}[Y|X]$, but I think that is not mathematically correct.