# CDF of sum of dependent random variables

Suppose that $X$ and $Y$ are $dependent$ random variables, what would be the cumulative distribution of $X+Y$?

That is, what is $P(X+Y\le c)$ for any integer c?

Note that we do not know their joint distribution

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The answer depends on the joint distribution of $X$ and $Y$. For any $\alpha$, not necessarily an integer, we have that for discrete random variables, $$P\{X+Y\leq \alpha\} = \sum \sum P\{X = u_i, Y = v_j\} = \sum \sum p_{X,Y}(u_i,v_j)$$ where the double sum is over all $i$ and $j$ such that $u_i + v_j \leq \alpha$. For jointly continuous random variables, we have $$P\{X+Y\leq \alpha\} = \int_{-\infty}^\infty \int_{v=-\infty}^{v=\alpha - u}f_{X,Y}(u, v)\,\mathrm dv\,\mathrm du.$$