Special case of Bertrand Paradox or just a mistake? I've been working on a question and it seems I have  obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in case anyone is interested.


I want to know what is the average distance between two points on a circle of radius 1 where we consider only the boundary points.

My attempt is as follows: 
Consider a segment of the diameter x which is uniformly distributed between 0 and 2. Then you can calculate the distance between the points (2,0) and the point determined by x just by elementary geometry as this picture shows:

Here in the picture, the green segment is the geometric mean and the orange one is the distance whose distribution we want to know.
Just by calculating the expected value, we obtain:
$E\left(\sqrt{(4-2X)}\right) = \int_{0}^{2} \sqrt{(4-2x)}\cdot\frac{1}{2} dx = 1.333.... = \frac{4}{3}$ 
Where $\sqrt{(4-2x)}$ is the transformation of the random variable and $\frac{1}{2}$ is the pdf of a uniform distribution $[0,2]$. 
Also, if we derive the pdf of the transformation we obtain the same result:
$y = \sqrt{(4-2x)} , x = 2- \frac{y^2}{2}, \mid\frac{d}{dy}x\mid = y$
$g(y)=f(x)\cdot\mid\frac{d}{dy}x\mid = \frac{1}{2}\cdot y$
$E(Y)= \int_{0}^{2}y\cdot\frac{1}{2}\cdot y dy = 1.333.... = \frac{4}{3} $
I have seen a different approach somewhere else where the distribution of the angle is considered as a uniform distribution between 0 and $\pi$ and the final result was:
$1.27... = \frac{4}{\pi}$
That's pretty much the problem I found. Maybe I just did it wrong in some step but it all makes sense to me. I know this is not exactly what we refer as Bertrand paradox but it just suggests something like that because both problems handle with segments in circumference and maybe my result is wrong because it does not hold for rotations of the circle or something like that (I read a little bit about the Bertrand's Paradox).

That's pretty much it. Also sorry for my bad English and maybe I'm also wrong in something pretty elemental since I've just started learning about probability theory. It's also my first post so I will try to improve my exposition and LateX use in the following ones.

Thanks, Erick Wong for your feedback. After your answer, I calculated the distribution of the arc length subject to the uniform distribution of the point on the diameter. In fact: if we want to express the arc length $l$ as a function of $x$, $l = f(x)$ we obtain: 
$l = \arccos(1-x), x = 1-\cos{l}, |\frac{d}{dl}(x)| = \sin{l}$
$l_{pdf} = x_{pdf} \cdot |\frac{d}{dl}x| = \frac{1}{2}\cdot \sin{l}$.
So the arc length does not distribute uniformly, we have "lost it", we might say. That's what was wrong.
For instance, if the arc length obeys a uniform distribution [0, $\pi$], then we can calculate the segment $s$ as a function of the arc length: 
We know $l = f(x)$ and want to know $s = h(l)$. If we calculate $s = g(x)$ we're done:
From the image on the question I post, $s = g(x)=\sqrt{2x}$ (or the opposite segment $\sqrt{4-2x}$ ) then $h = g \circ f^{-1}$, $s = \sqrt{2(1-cosl)}$
$E(s) = \int_0^\pi{\sqrt{2(1-cosl)}\frac{1}{\pi}dl}=1.273... = \frac{4}{\pi}$
Also the pdf:
$s = h(l) = \sqrt{2(1-cosl)} ,  l=h^{-1}(s)= 2\cdot \arcsin(\frac{s}{2}), |\frac{d}{ds}h^{-1}|=\frac{2}{\sqrt{4-s^2}}$
$s_{pdf} = \frac{1}{\pi} \frac{2}{\sqrt{4-s^2}}$
$E(s) = \int_0^2{s\cdot \frac{1}{\pi} \frac{2}{\sqrt{4-s^2}}ds} = 1.273... = \frac{4}{\pi}$
So we're done. Also, the pdf of the segment suggests something related to the Cauchy distribution. Not exactly but certainly it has to do something with it. If we read the description of Cauchy distribution in Wolfram MathWorld:
"The Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution, is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis."
And that's it. A really fascinating problem that introduces some subtle ideas of probability theory. If anyone knows something else please give me feedback. I really think there's a nice connection with Cauchy distribution.
