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.