- #1
rabbed
- 243
- 3
Hi
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
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