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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 $P\{X+Y\leq \alpha\} = \int_{-\infty}^\infty \int_{v=-\infty}^{v=\alpha - u}f_{X,Y}(u, v)\,\mathrm dv\,\mathrm du.$

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    The solution Dilip provided can also be expressed in terms of the conditional probability, which is sometimes attainable in terms of the problem. Of course, if you have this, you can likely find the joint distribution... But the intermediate step is convenient in some cases.2019-01-24