Is there a way to "see" that ∞∑r=0∞∑x=r+1P(X=x)=∞∑x=1x−1∑r=0P(X=x)? Thanks.
Answer
It doesn't matter what you sum (as long as the sums are convergent). The points in the (r,x) plane that are being summed over can be illustrated by a diagram like this:
x
5 *****
4 ****
3 ***
2 **
1 *
012345 r
The two sides in the identity correspond to summing by columns first or rows first.
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