<s>[INST] How to find the sum of 2 discrete random variables Let X and Y be independent random discrete variables and 
$X = \begin{pmatrix}-1&0&1\\\frac13&\frac13&\frac13\end{pmatrix}$ and $Y = \begin{pmatrix}0&1\\\frac13&\frac23\end{pmatrix}$
Then what is $X + Y$? As a discrete random variable.. I understood that I have to add the first row just like in matrix addition but what happens when they don't have the same size and how to add the probabilities? Also what is the difference between independent variables and incompatible variables?
The missing entries were:
$X + Y = \begin{pmatrix}-1&0&1&2\\\frac19&\frac39&\frac39&\frac29\end{pmatrix}$
 [/INST] Hint: Can you enumerate all the cases? For example, if $X+Y=1$ it means either $X=0$, $Y=1$ (probability $\frac{1}{3}\cdot\frac{2}{3}=\frac{2}{9}$) or $X=1$, $Y=0$ (probability $\frac{1}{3}\cdot\frac{1}{3}=\frac{1}{9}$) $\Longrightarrow P\left(X+Y=1\right)=\frac{2}{9}+\frac{1}{9}=\frac{1}{3}$.
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