Lets say I have $f_{X,Y}(x,y)$, as my continuous probability density function.
I want the $\Bbb P(0<y<1 \mid x=2 )$. I'm going to take the plane $x=2$ and intersect it with $f_{X,Y}(x,y)$, get a line/curve that is a continuous probability density function in terms of $y$.
Then I will integrate using the limits I want, wrt to $y$. Why is this wrong?
I don't want to use $f_{Y\mid X}(y\mid x) = f_{Y,X}(y,x)/f_X(x)$, since I don't like how at the end we can plug in a single value for $x$, in a density function.
Thank you