I Derive Convolution Expression for Z_PDF(z)

[Stephen Tashi]

As far as I can tell, you aren't presenting any logical arguments. You are conjecturing various formulas and asking for criticism of them. That's a permissible approach in the early stages of an investigation, but you should follow-up a conjecture by testing it with some simple examples instead of relying on my comments. it's ok to make conjectures by resorting to "magic" - such as writing down symbols like "dx/dx" without asking what they symbolize. But you should proceed to working specific examples that force you to make specific interpretations. (I'm about to get busy for a few days with the jobs of being executor of an estate, so I'm not going to have time to criticize a hundred different conjectures.) - In fact I just got a phone call and I must leave right now.

 

[rabbed]

I think the essence here is that for multiple variables/dimensions we need to use line/surface integrals
and that it starts to make more sense to use the joint PDF of the source RV's.
XY_PDF(x,y)*sqrt(dx^2 + dy^2) = X_PDF(x)*|dx| * Y_PDF(y)*sqrt(1 + (dy/dx)^2)

Hm, maybe regular multidimensional integration comes in when considering inequalities, like Z < X+Y.

I'll keep exploring. Thanks for all help!