Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$
What are the different methods to numerically compute the distribution X+Y, X*Y, X/Y, min(X,Y) etc...?
I read about Mellin transform and Monte-Carlo simulation but it seemed to me that since these methods go back to a long time ago, there must be something that already exists for such operations within a library or a module on a programming language like Matlab or R (or any other platform)
Any ideas|suggestions on this matter would be greatly appreciated!
If $X,Y$ are independent and have distribution function $F_X,F_Y$ and densities $f_X$,$f_Y$, you have $$\begin{eqnarray} &P(X+Y \leq z) &=& \int_{x+y \leq z} f_X(x) f_Y(y) \,d(x,y) = \int_{-\infty}^\infty \int_{-\infty}^{z-x} f_X(x) f_Y(y) \,dy \,dx \\ &&=& \int_{-\infty}^\infty f_X(x) F_Y(z-x) \,dx \text{,} \\ &f_{X+Y}(z) &=& \frac{d}{dz}P(X+Y \leq z) = \int_{-\infty}^\infty f_X(x) f_Y(z-x)\,dx \\ \text{ and } \\ \\ &P(\min\{X,Y\} \leq z) &=& P(X \leq z \text{ or } Y \leq z) = F_X(z) + F_Y(z) - F_X(z)F_Y(z) \text{,} \\ &f_{\min X,Y}(z) &=& f_X(z) + f_Y(z) - f_X(z)F_Y(z) - F_X(z)f_Y(z) \\\text{ respectively } \\ \\ &P(\max\{X,Y\} \leq z) &=& F_X(z)F_Y(z) \text{,} \\ &f_{\max X,Y}(z) &=& f_X(z)F_Y(z) + F_X(z)f_Y(z) \text{.} \end{eqnarray}$$ For $XY$ and $X/Y$ there are similar transformation formulas, and you can always evaluate the resulting integrals numerically. For $X+Y$, using that $f_{X+Y}$ is the convolution of $f_X$ and $f_Y$ might help.