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Ok, so randomizing three random variables, X, Y and Z, each from a standard normal distribution, then plotting these in an ordinary cartesian coordinate system gets me a spherically symmetric cloud of points.

Now I want to create this cloud having the same probability distribution but by using a sphere formula having a uniform direction distribution and a radius of some function f(R) where R has some unknown distribution:

X = f(R)*sqrt(1-V^2)*sin(O)

Y = f(R)*V

Z = -f(R)*sqrt(1-V^2)*cos(O)

|J| = | d(XYZ) / d(OVR) | = f(r)^2*f'(r)

O_PDF(o) = 1/(2*pi) where (0 < o < 2*pi)

V_PDF(v) = 1/2 where (-1 < v < 1)

R_PDF(r) = ?

So I want to calculate R_PDF(r). I know that R^2 = X^2 + Y^2 + Z^2 so I use the CDF method to first

find the distribution of XS = X^2 knowing that X_PDF(x) = 1/sqrt(2*pi)*e^(-x^2/2).

This gets me XS_PDF(xs) = 1/sqrt(2*pi*xs)*e^(-xs/2) which is the same for YS_PDF and ZS_PDF

Then I want to calculate RS_PDF(rs) from RS = XS + YS + ZS, knowing XS_PDF(xs), YS_PDF(ys) and ZS_PDF(zs):

RS_PDF(rs) =

RS_CDF'(rs) =

( P(RS < rs) )' =

( P(XS + YS + ZS < rs) )' = ?

What to do here?

Now I want to create this cloud having the same probability distribution but by using a sphere formula having a uniform direction distribution and a radius of some function f(R) where R has some unknown distribution:

X = f(R)*sqrt(1-V^2)*sin(O)

Y = f(R)*V

Z = -f(R)*sqrt(1-V^2)*cos(O)

|J| = | d(XYZ) / d(OVR) | = f(r)^2*f'(r)

O_PDF(o) = 1/(2*pi) where (0 < o < 2*pi)

V_PDF(v) = 1/2 where (-1 < v < 1)

R_PDF(r) = ?

So I want to calculate R_PDF(r). I know that R^2 = X^2 + Y^2 + Z^2 so I use the CDF method to first

find the distribution of XS = X^2 knowing that X_PDF(x) = 1/sqrt(2*pi)*e^(-x^2/2).

This gets me XS_PDF(xs) = 1/sqrt(2*pi*xs)*e^(-xs/2) which is the same for YS_PDF and ZS_PDF

Then I want to calculate RS_PDF(rs) from RS = XS + YS + ZS, knowing XS_PDF(xs), YS_PDF(ys) and ZS_PDF(zs):

RS_PDF(rs) =

RS_CDF'(rs) =

( P(RS < rs) )' =

( P(XS + YS + ZS < rs) )' = ?

What to do here?

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