I am working with the TransformedDistribution function, and have successfully derived PDF functions for r, theta and phi.
When I try and calculate the PDF for z = r * Cos[theta] via the TransformedDistribution function, I get stuck. My gut tells me that there is an assumption missing, but I am not sure.
I need the PDF for the interval 0 < t < 2.
My work is attached. Any suggestions?
Derivation.nb (14.2 KB)
You must work with the distribution, not with the PDF.
Following Gianluca Gorni’s suggestion:
disttheta = TransformedDistribution[ArcCos[2*v - 1], v \[Distributed] UniformDistribution[{0, 1}]]
distr = TransformedDistribution[v^(1/3), v \[Distributed] UniformDistribution[{0, 1}]]
distphi = TransformedDistribution[v, v \[Distributed] UniformDistribution[{0, 2*Pi}]]
distz = TransformedDistribution[u*Cos[v], {u \[Distributed] distr, v \[Distributed] disttheta}]
PDF[distz, z]
I understand. Thank you, Gianluca
Yes, Jim. That works for me. Thank you.