The joint probability density function of $x$ and $Y$ is given by

$f(x,y)=x{e}^{-x(y+1)},x>$

$0,y>0$

Which of the following statements are correct?

Check as many boxes as applicable.

The conditional variance of $Y$ given $x=x$ for $x>0$ is $\frac{1}{x}.$

The conditional mean of $Y$ given $x=x$ for $x>0$ is $\frac{1}{x}.$