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There are so many different kinds of probability distributions regarding a uniform distribution

of points on the surface or inside a hypersphere in 2D and 3D and it's hard to see the big picture

or any pattern between them. I'm confused and exhausted :)

The overall fuzzy plan is to go from 2D polar coordinates (integrating by r*dr*dθ) to 3D suitable

coordinates (integrating by whatever the equivalent might be).

In both cases, I want to multiply the independent components PDF's, get the CDF and invert it

to get the cartesian coordinates of a point that will be uniformly distributed inside the

hypersphere. I guess normalization after this will give me a random vector?

Then I want to show that independent standard normal distributed coordinates will also accomplish

this after normalization, in a more unified way.

Does that plan make sense from a mathematician point of view and have I got something wrong?

Right now I'm stuck at the 2D case, I guess I want to multiply V_PDF(v) = 1/(2*pi) with some R_PDF(r).

How do I get such an R_PDF(r)?

I should end up with (x,y) = r*sqrt(u)*e^(i*2*pi*v) with u and v ~ U(0,1).

/rabbed

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# Random vectors

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