Perhaps you need the algebra of random variables. By using this algebras and the standard techniques of calculus you can, at least in principle, determine the PDF of functions $f(X_1, \dots, X_n)$ of $n$ random variables from their PDF: this of course includes the standard difference and sum of random variables but also the product and quotient. A now classical text is [1] which, moreover, is available at the Internet Archive (with some restrictions).

**Reference**

[1] Melvin Dale Springer, *[The algebra of random variables](https://archive.org/details/algebraofrandomv0000spri/page/n5/mode/2up)* (English) 
Wiley Series in Probability and mathematical Statistics. New York etc.: John Wiley & Sons, pp. XIX+470 (1979), ISBN:0-471-01406-0, [MR519342](https://mathscinet.ams.org/mathscinet-getitem?mr=519342), [Zbl 0399.60002](https://zbmath.org/0399.60002).