I have some data set where
X: time spent using Feature 1
Y: time spent using Feature 2 (a subfeature of Feature 1)
Note that the two variables are always non-negative, and also, Feature 2 is located inside of Feature 1, so if I use Feature 2, I must be using Feature 1 (i.e X>Y), so the two are not independent at all.
EDIT: See comment below. The data can come as either paired or unpaired, whichever is easier.
I want to estimate E[Y], but I only have the arbitrary moments for X|Y=0, X|Y>0, and X.
How can I estimate E[Y]?
I would start with the conditional expectation formula
$f_{X|Y} (x|y) =\frac{f_{X,Y} (x, y)}{f_Y (y)}$
As I said, I can estimate both $f_{X|Y} (x|y=0)$ and $f_{X|Y} (x|y>0)$ with the data I have.
But I don't think I can estimate $f_{X,Y} (x, y)$, because to get $P(X \le x, Y \le y)$ would require (I think) knowing that $Y \le y$ but i have no idea about the length of $Y$ in the first place.
Does this all make sense? How can I proceed? What additional info would I need to estimate $f_{X,Y} (x, y)$? I still haven't used the fact that I know $f_X(x)$.