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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1 Answer
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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 $$P\{X+Y\leq \alpha\} = \int_{-\infty}^\infty \int_{v=-\infty}^{v=\alpha - u}f_{X,Y}(u, v)\,\mathrm dv\,\mathrm du.$$
answered Nov 18, 2012 at 22:09
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$\begingroup$ thanks I should have mentioned that we do not know their joint distribution. $\endgroup$– MayNov 18, 2012 at 23:08
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3$\begingroup$ @May If you do not know the joint distribution, then the problem is not solvable. It might be possible to obtain some bounds in some cases. $\endgroup$ Nov 19, 2012 at 2:38
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$\begingroup$ 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. $\endgroup$– knrumseyFeb 3, 2017 at 20:22