- #1
Dante Tufano
- 34
- 0
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
##\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