# Can't find this sigma notation syntax: $\sum_{y:p(x,y) > 0}p(x, y)$

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.

• 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. yesterday

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...
• 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 $\sum_{y=0}^4 p(2,y)$? yesterday
• or $\sum_{y=1}^{4}$ ... since you don't have a y=0 ...