Suppose that the joint distribution of $X$ and $Y$ is uniform over the region in the $xy$-plane bounded by $x=-1,x=1,y=x+1, \text{ and }y=x-1$.
What is $\mathbb{P}(XY>0)$?
What is the conditional p.d.f. of $Y$ given that $X=x$?
Suppose that the joint distribution of $X$ and $Y$ is uniform over the region in the $xy$-plane bounded by $x=-1,x=1,y=x+1, \text{ and }y=x-1$.
What is $\mathbb{P}(XY>0)$?
What is the conditional p.d.f. of $Y$ given that $X=x$?
The region in the $XY$-plane is as shown below.
HINT for the first part. Identify the regions where $XY > 0$. And integrate over the region to get $\mathbb{P}(XY > 0)$.
HINT for the second part. Recall that $f_{Y|X=x} = \dfrac{f_{XY}}{f_X}$, where $f_X = \displaystyle \int_y f_{XY} dy$.