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

  1. Feb 21, 2015 #1
    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
     
  2. jcsd
  3. Feb 21, 2015 #2

    vela

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  4. Feb 21, 2015 #3
    So to do this, I would have to develop a general expression for the n-dimensional Jacobian?
     
  5. Feb 21, 2015 #4

    vela

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    Yeah, pretty much. Check out that Wikipedia page.
     
  6. Feb 21, 2015 #5
    Okay, I think that might be all the direction I needed! Thanks. I am running through some calculations and will update soon.
     
  7. Feb 21, 2015 #6
    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.
     
    Last edited: Feb 21, 2015
  8. Feb 22, 2015 #7
    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?
     
  9. Feb 22, 2015 #8
    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.
     
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