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I'm learning probability and came across a sigma notation that I don't know what that means.

$X$ and $Y$ are both discrete random variables and their their joint probability function is: $$p(x,y) = P\{X = x, Y = y\}$$
from $p(x,y)$ we can get the probability function of $X$ like this:

$$p_x(x) = P\{X = x\} = \sum_{y:p(x,y) > 0}p(x, y)$$
same for $y$, but instead of '$y:$', it's '$x:$'.

As you can see there is no top (im guessing it means sum to infinite?), and about the bottom part I'm completely lost.

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    $\begingroup$ Yes. That is horrendous notation. Clearly $x,y$ must be discrete. They could be infinite. From the bottom part you can drop $p(x,y)>0$ because adding the terms with $p(x,y)=0$ to the sum won't change it. $\endgroup$
    – Kurt G.
    yesterday

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The : can be read as something like "such that". So the terminology specifies the sum over all $y$ for which $P(x, y) > 0$. As Kurt G. says, it is somewhat redundant in this case...

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  • $\begingroup$ so if say I have X = 1,2,3 and Y = 1,2,3,4 and I want to find $p_x(2)$ it will be the same as writing $\sum_{y=0}^4 p(2,y)$? $\endgroup$
    – Ellie
    yesterday
  • $\begingroup$ or $\sum_{y=1}^{4}$ ... since you don't have a y=0 ... $\endgroup$
    – DrM
    yesterday
  • $\begingroup$ yes my mistake, thanks. $\endgroup$
    – Ellie
    yesterday

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