Is this a property of summations?

Question

My main problem comes from understanding the process from line 2 to line 3. Is that a property of summations that im not aware of? it says in the script that this is a "useful trick".

i can't understand it as a procedure that one performs on the second line to turn it into the third line. I only have a basic understanding of it to be true in terms of the second summation serving to sum P(X=y) (y-1) times.

Answer 1

He's just interchanging the order of summation. In line two, for each $x$, you sum over all $y>x$. In line 3, for each y you sum over all $x<y$. But $x<y$ and $y<x$ are equivalent, so you're summing over the same $(x,y)$ pairs in both cases.

EDIT I originally reversed "row" and "column" in this explanation. See the comments.

If you imagine marking the pairs $(x,y)$ in the plane that satisfy these conditions with dots, you get an infinite "triangle" of dots. In the first case,you sum over the columns, which is inconvenient because they have infinitely many dots, and in the second case, you sum over the rows, which have only finitely many dots, so it's a lot easier.

Answer 2

The following is valid if all summands are non-negative: \begin{align*} \sum_{x=0}^\infty\sum_{y=x+1}^\infty f(x,y) \color{blue}{=\sum_{0\leq x<y<\infty}f(x,y)=} \sum_{y=1}^\infty\sum_{x=0}^{y-1}f(x,y) \end{align*}

The middle sum might help to see the connection between line $2$ and line $3$.