What is the expected distance between these two points? Suppose you have a straight line of length L, and a point is chosen at random along the line.  

Now suppose a second point is chosen at random to the left of the first point.

What is the expected distance between these points?

Note:  I am aware that if two points are independently chosen at random, the expected distance is L/3, but is it the case in this scenario where the second point is dependent upon the first?  Please justify answers with mathematics.
We have the density function for uniform $x\in[0,L]$ and uniform $y\in[0,x]$
$$
f(x,y)=\frac 1L\cdot\frac 1x
$$
so it is
$$
\int_0^L\int_0^x (x-y)\frac 1L\cdot\frac 1x\ dy\ dx=\frac L4
$$
which can be confirmed by this Wolfram|Alpha-computation.
let [0, L] represent the line.

If X1 and X2 are the Random Variables representing the expected positions of the two points, then:

E(X1 - X2) = E(X1) - E(X2) = L/2 - L/4 = L/4