I have two dependent continuous random variables (RVs) $X$ and $Y$ and I'm interested in determining the CDF of the sum, i.e., $F_{X+Y}(t) = \mathbb{P}(X+Y \leq t)$. I know the marginal of $X$ and $Y$ ($f_X(\cdot)$ and $f_Y(\cdot)$, respectively). Using the copulas framework, I know that the joint distribution is given by $$ f(x,y) = c(F_X(x),F_Y(y)) \times f_X(x) f_Y(y).$$ Thus, the CDF can be written as $$F_{X+Y}(t) = \int \int \mathbb{1}_{(x+y\leq t)} c(F_X(x),F_Y(y)) \times f_X(x) f_Y(y) dx dy,$$ where $\mathbb{1}_{(\cdot)}$ is the indicator function. At this stage, even for the simplest case, I feel that computing the above in closed-form seems difficult, so instead, I can write $$F_{X+Y}(t) = \mathbb{E}[\mathbb{1}_{(X+Y\leq t)} c(F_X(X),F_Y(Y))]$$ and use Monte Carlo simulations to evaluate the above.
The context is $X$ and $Y$ are Rayleigh RVs, I know that they are correlated but I don't have their joint distributions (I have access to samples from the joint distribution but not the analytical form).
Provided that my argument is correct, I have the following questions:
- How to choose the copulas $C$?
- How to sample from $X$ and $Y$ when they are dependent? Thanks!