It's likely some variation of this has already been answered, but I'm unsure the correct terminology to do proper searching for it. If there is, sorry in advance. In the same regard, I'm not sure how to pose this generally, so I'll just ask based on my specific scenario.
I'm looking at energy deposition data from an alpha-source implanted in a material. In short, it's monoenergetic, but because of various emission angles, the outcoming energy has a tailed-distribution I'd like to model (so I can fit binned data).
Because of isotropic emission, the azimuthal angle is NOT uniformly distributed. From a surface area integral of a unit-sphere in 3D, I believe I'm just in saying the number of emissions in a $d\theta$ band (from $\theta$ to $\theta + d\theta$) is proportional to $\cos(\theta) - \cos(\theta+d\theta)$. I.e. $dN \propto \sin(\theta)d\theta$.
Making very convenient assumptions of no path-deflections and a constant stopping power, the energy loss in the material should be a function of the azimuthal/emission angle, $\Delta E \propto \frac{1}{\cos(\theta)}$.
To get an idea of a produced spectrum, I ran a simple script that produces weights and energy losses at many angles, and binned it. There's a screenshot at the bottom for clarity. The result makes intuitive sense; a very large portion are hardly hindered, but there is a tail of increased energy loss.
I want to write a function for the PDF given the provided theory. I suppose the more general question would then be: Given a function $f(\theta)$ that depends on a non-uniformly distributed variable $\theta$, how does one derive the resulting PDF for $f$?
Note, in my case the PDF function for $\Delta E$ is a distribution of energy not angle.
