Finding distribution of $y$ intercept given by line generated by two uniformly chosen points in $[0,1]^2$ The method I'm using seems super inefficient. What I did was define $4$ RVs namely $X_1,X_2,Y_1,Y_2$ and thus my two uniformly random points are $(X_1,Y_1),(X_2,Y_2)$ and hence the $y$ intercept is $-X_1(Y_2-Y_1)/(X_2-X_1)+Y_1$ but them I'll have to do Jacobian tranform multiple times. 
Is there any simpler method I seem to be missing?

Empirically the density and cumulative distribution functions seem to look something like this, with the probability of the intercept being in $[0,1]$ being $\frac12$ and the density being constant in that interval

I am not convinced that this distribution has a mean, though it is obviously centred at $\frac12$; even if it does, I would be very surprised if it has a finite second moment 
