Probability distribution, a 1-D dirac delta in n-dimensions

[Dante Tufano]

Hey everybody, I'm an engineering Ph.D. so my knowledge of n-dimensional Euclidean spaces is lacking to say the least. I'm wondering what sort of approach I can take to solve this problem.

$\boldsymbol{1.}$ and $\boldsymbol{ 2. }$

I am given a probability distribution for a random variable:

$\boldsymbol{X}=(X_1,X_2,...,X_N)\in \mathbb{R}$

which is concentrated on the n-1 dimensional sphere of radius R

$\Omega\equiv\{\boldsymbol{x}\in\mathbb{R}^n:\lvert \boldsymbol{x} \rvert^2=\sum_{j=1}^n x^2_j=R^2 \}$

I am given a probability distribution

$p(\boldsymbol{x})=\dfrac{n}{A_{n-1}}\delta(\lvert{\boldsymbol{x}}\rvert^n-R^n)$

where $A_n=\dfrac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)}$ is the area of an n-dimensional unit sphere on $\mathbb{R}^{n+1}$

and have to prove that

$\int p(\boldsymbol{x})\mathrm{d}\boldsymbol{x}=1$

My professor told me there is some sort of trick to solving the problem. I figure I have to somehow match the dimensions of this one-dimensional dirac delta function to those of the integral over #\mathbb{R}^n#, but I'm not really sure how to approach this. I've been trying multiple ways of manipulating the dirac delta function, but I'm pretty stumped.

$\boldsymbol{3.}$ It's fairly obvious that we have

$\int p(\boldsymbol{x})\mathrm{d}\boldsymbol{x}=\dfrac{n}{A_{n-1}}\int_{x_n}...\int_{x_2}\int_{x_1}\delta((x_1^2+x_2^2+...+x_n^2)^{n/2}-R^n)\mathrm{d}x_1\mathrm{d}x_2...\mathrm{d}x_n$

but I am not sure where to go from here

 

[vela]

I'd try changing to spherical coordinates.

http://en.wikipedia.org/wiki/N-sphere

 

[Dante Tufano]

So to do this, I would have to develop a general expression for the n-dimensional Jacobian?

 

[vela]

Yeah, pretty much. Check out that Wikipedia page.

 

[Dante Tufano]

Okay, I think that might be all the direction I needed! Thanks. I am running through some calculations and will update soon.

 

[Dante Tufano]

Okay, so I now have the integral

$\int_{\mathrm{d}^nV}\delta(r^n-R^n)\mathrm{d}^nV=\int_{\phi_{n-1}=0}^{2\pi}...\int_{\phi_1=0}^{\pi}\int_{r=0}^{R}\delta(r^n-R^n)r^{n-1}\sin^{n-2}(\phi_1)...\sin(\phi_{n-2})\mathrm{d}r\mathrm{d}\phi_1...\mathrm{d}\phi_{n-1}$

I know I am really close. What is the trick to manipulating the $\delta(r^n-R^n)$ term here? I am not sure how to deal with the power of $n$ on $r$.

Edit: I have figured out my problem. I need to use the equation for manipulating a delta function with only one root. All the rest are imaginary. Thanks for the help.

 

[Dante Tufano]

Okay, so now that I've done this. How would I go about calculating the marginal probability of one of the events $X_j$? I know that typically for a marginal probability, we integrate as such:

$p_X(x)=\int{-\infty}^{\infty}p_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y$

So clearly, I have to extend this integral to the case of an n-dimensional random vector, in which case I integrate over every component of $\boldsymbol{X}$ other than some component $X_j$. How do I accomplish this in spherical coordinates though? If that's not the right question, then maybe the right question is how to handle this very peculiar delta function in Cartesian coordinates?

 

[Dante Tufano]

I've been thinking about the problem, and am only getting more confused. Once again, I don't have a math background so I'm not sure if there are some analogues between lower dimensional space and n-space that might help me gain some intuition about this problem.