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.

[alThor]

Hints only:

You have (a) right.

There is an easier way to do (b). What does common sense tell you the answer is? (If you can't figure out the easier way, you could still use your method and figure out E(XY_bar * YX_bar) using integrals.)

(c) is more difficult. To get a common-sense answer, suppose p isn't fixed, but suppose you fix X first, instead. Then randomly draw Y. Given Y, finally, draw p randomly. Do you think p will end up being in XY or YX? How likely are those two outcomes, relative to each other? After you figure that out, draw the density function 'f'. Calculate the cumulative 'F' only afterwards.

alThor

[peterchi]

[ QUOTE ]
Hints only:

You have (a) right.

There is an easier way to do (b). What does common sense tell you the answer is? (If you can't figure out the easier way, you could still use your method and figure out E(XY_bar * YX_bar) using integrals.)

(c) is more difficult. To get a common-sense answer, suppose p isn't fixed, but suppose you fix X first, instead. Then randomly draw Y. Given Y, finally, draw p randomly. Do you think p will end up being in XY or YX? How likely are those two outcomes, relative to each other? After you figure that out, draw the density function 'f'. Calculate the cumulative 'F' only afterwards.

alThor

[/ QUOTE ]
Firstly, thank you!

For part b:
Okay, after thinking about it, common sense tells me that the correlation is -1, since they have a decreasing linear relationship... right? But how can I formalize this into a coherent answer?

For part c:
okay that kind of makes sense.

Would I want to do something like
P(p is in XY)*XY_bar + P(p is in YX)*YX_bar?

Thanks so much again.

[alThor]

[ QUOTE ]
For part b:
Okay, after thinking about it, common sense tells me that the correlation is -1, since they have a decreasing linear relationship... right? But how can I formalize this into a coherent answer?

[/ QUOTE ]

Write that relationship down! Replace for YX in the covariance (or corr.) formula, and just do the algebra from there. "Substitute and solve" to get the answer you think is right.

[ QUOTE ]
For part c:
okay that kind of makes sense.

Would I want to do something like
P(p is in XY)*XY_bar + P(p is in YX)*YX_bar?


[/ QUOTE ]

That looks like you are trying to compute an expected value of some kind, so it's not quite right. If you prefer to go directly to the cumulative F(), ask yourself, before drawing Y and p, what is the probability that p will land in an arc sized less than half the circle? (There is always an arc sized less than (or equal) half the circle, but what is the probability p will end up in it, before knowing the exact size of that smaller arc? You can figure that out with an integral.)

alThor