Random points on a circle

[peterchi]

Another homework question for you brilliant people out there. I got as far as I could and now I'm stuck. Any help is greatly appreciated!

Assume that you have a circle with a point P designated on the perimeter. You choose two points X and Y along the perimeter. Let XY_bar represent the length of the arc from X and Y (drawn clockwise from X) and let YX_bar represent the length of the arc from Y to X (drawn clockwise from Y). Suppose the radius of the circle is r.
a) What are the marginal distributions of XY_bar and YX_bar?
b) What is the correlation between XY_bar and YX_bar?
c) What is the distribution of the length of the arc that covers the point P?


Part a:
XY_bar ~ U(0, 2*pi*r)

so...

Fxy_bar(xy_bar) =
0 for xy_bar < 0
xy_bar/(2*pi*r) for 0 < xy_bar < 2*pi*r
1 for xy_bar > 2*pi*r

And then the same thing for YX_bar.

How am I doing so far?



Part b:
Corr(xy_bar, yx_bar) = Cov(xy_bar, yx_bar) / [sqrt(Var(xy_bar) * sqrt(Var(yx_bar)]

so

Cov(xy_bar, yx_bar) = E[(XY_bar - E(XY_bar))(YX_bar - E(YX_bar))]

= E[XY_bar * YX_bar] - E(XY_bar)E(YX_bar)

so...

E(XY_bar) = E(YX_bar) = pi*r, right?

But now what is E(XY_bar * YX_bar) ???

And then for the Variances, I know that
Var(X) = E(X^2) - mu^2
but I'm not sure how to use that here.


Part c:
I'm not really sure how to approach this part. I know that it is more likely to be in the longer segment, due to length-biased sampling. But I'm not sure how that affects the distribution.


Many many thanks in advance.