Saturday, 23 February 2013

combinatorics - Double summation limits



Is there a way to "see" that r=0x=r+1P(X=x)=x=1x1r=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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