Basically, I'm trying to find what the mean and standard deviation of the distance between two random "hyperpoints" from a 512 dimensional hypercube of size 2 would be.
I was thinking the mean would be $\displaystyle{\int_{-1}^{1}\int_{-1}^{1}...\int_{-1}^{1}\sqrt{(x_{1,1}-x_{1,2})^2+(x_{2,1}-x_{2,2})^2+(x_{3,1}-x_{3,2})^2+...+(x_{512,1}-x_{512,2})^2}}dx_{1,1}dx_{1,2}dx_{2,1}dx_{2,2}...dx_{512,1}dx_{512,2}$
A. Does that seem right? B. If so, is there an analytic solution, and C. If not.. I estimated the mean as 18.4691104453 and the standard deviation as 0.483367889827 by simply choosing a million random 512 dimensional points in that range, and calculating a sample mean and sample standard deviation from that. Does that seem valid?
What I'd really like is a closed form solution for an n-dimensional hypercube...
Thanks!