 Solution

The marginal probability density function is $f_Y(y)={\int }_{\mathbb{R}}{f}_{X,Y}(x,y)dx$. For $y$ with $f_Y(y)>0$, the conditional density isIf the conditional first absolute moment is finite, the conditional expectation isConditioning here uses a density, since the individual event $Y=y$ normally has probability zero; a conditional law is specified only up to marginal-null values of $y$.