N-dimensional broken stick problem -- find joint density

[psie]

Homework Statement

A point ##P## is chosen uniformly in an ##n##-dimensional ball of radius ##1##. Next, a point ##Q## is chosen uniformly within the concentric sphere, cantered at the origin, going through ##P##. Let ##X## and ##Y## be the distances of ##P## and ##Q##, respectively, to the common center. Find the joint density function of ##X## and ##Y## and the conditional expectations ##E(Y\mid X=x)## and ##E(X\mid Y=y)##.

Relevant Equations

I am not sure.

There are also two hints, which I will share with you now. The first hint says to start with the case ##n=2##. I've drawn a unit disc and a circle inside this unit disc, but I do not know how to reason further.

The second hint says that the volume of an ##n##-dimensional ball of radius ##r## is equal to ##c_nr^n##, where ##c_n## is some constant, and that this is of no interest to the problem. Somewhere this makes sense as we are only concerned with distances.

Then there's also a remark to the problem, namely that for ##n=1##, we rediscover the broken stick problem.

I'd be grateful for any help on this problem. The answer for the joint distribution should be ##f_{X,Y}(x,y)=n^2\frac{y^{n-1}}{x^n}## for ##0<y<x<1## (I also have the answer for the conditional expectations, if anyone's interested). But how to obtain these answers I have yet to understand.

 

[Orodruin]

Start with finding the radial distribution for a uniformly sampled sphere and go from there as you would in the broken stick problem. The only difference is that you are not breaking the stick uniformly.

 

[psie]

Ok. There's actually an example in the book where the author determines the radial distribution of a dart board for a beginning dart player (i.e. the darts are assumed to land uniformly on the dart board). The author gets that the radial distribution of ##R##, the distance from the origin, has density ##f_R(r)=2r## for ##0<r<1##. So in this example I believe we simply have ##X=R##. I don't see yet how this can help me determine the joint distribution. Moreover, I think the density of ##Y## is a bit trickier and I'm not sure how to obtain it.

 

[Orodruin]

You have the conditional probability ##f_{Y|X=x}## and you have ##f_X##. From this you shoukd be able to obtain the joint pdf.