I’m working on a physics problem where we have a measured photon energy spectrum (I’m thinking of it as a probability mass function, PMF), which is created by an energy spectrum of electrons which impact the atmosphere. The two spectra are related by a (known) matrix multiplication.

If it’s valid to think about the two spectra as PMFs, is there a relation between the first few moments of the two distributions?

I’m asking because the matrix is very ill-conditioned, and the first few moments of the resulting spectrum is all I need to know for the physics problem. The “full problem” of finding the inverse of the ill-conditioned matrix is handled through regularization (Tikhonov, LASSO, etc.). Is it possible to pose a better conditioned problem, by seeking “less information” about the transformed distribution?

I did some searching and found this: Characteristic Function and Random Variable Transformation. I think that the result there pertains to individual bins in my measured spectrum being considered together as a vector random variable. I can, of course, get the expectation value and the variance of each bin, but what I’m looking for is a way to get the center and approximate width of the resulting distribution, without solving the full inverse problem.