Poisson race distribution/CalcTable1

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[ilmath]k[/ilmath] [ilmath]X[/ilmath] [ilmath]Y[/ilmath]
[ilmath]0[/ilmath] [ilmath]X\eq 0[/ilmath] [ilmath]Y\eq 0[/ilmath]
[ilmath]X\eq 1[/ilmath] [ilmath]Y\eq 1[/ilmath]
[ilmath]X\eq 2[/ilmath] [ilmath]Y\eq 2[/ilmath]
[ilmath]\vdots[/ilmath] [ilmath]\vdots[/ilmath]
[ilmath]X\eq i[/ilmath] [ilmath]Y\eq i[/ilmath]
[ilmath]1[/ilmath] [ilmath]X\eq 1[/ilmath] [ilmath]Y\eq 0[/ilmath]
[ilmath]X\eq 2[/ilmath] [ilmath]Y\eq 1[/ilmath]
[ilmath]X\eq 3[/ilmath] [ilmath]Y\eq 2[/ilmath]
[ilmath]\vdots[/ilmath] [ilmath]\vdots[/ilmath]
[ilmath]X\eq i+1[/ilmath] [ilmath]Y\eq i[/ilmath]
[ilmath]2[/ilmath] [ilmath]X\eq 2[/ilmath] [ilmath]Y\eq 0[/ilmath]
[ilmath]X\eq 3[/ilmath] [ilmath]Y\eq 1[/ilmath]
[ilmath]X\eq 4[/ilmath] [ilmath]Y\eq 2[/ilmath]
[ilmath]\vdots[/ilmath] [ilmath]\vdots[/ilmath]
[ilmath]X\eq i+2[/ilmath] [ilmath]Y\eq i[/ilmath]
[ilmath]\cdots[/ilmath]
[ilmath]j[/ilmath] [ilmath]X\eq j[/ilmath] [ilmath]Y\eq 0[/ilmath]
[ilmath]X\eq j+1[/ilmath] [ilmath]Y\eq 1[/ilmath]
[ilmath]X\eq j+2[/ilmath] [ilmath]Y\eq 2[/ilmath]
[ilmath]\vdots[/ilmath] [ilmath]\vdots[/ilmath]
[ilmath]X\eq i+j[/ilmath] [ilmath]Y\eq i[/ilmath]

This shows why we sum over what we sum over!

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