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?