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## Homework Statement

[tex]\sum\limits_{j=0}^\infty \binom{j}{r} p^r (1-p)^{j-r} (1-q) q^j[/tex]

where p and q are between 0 and 1, and r is a positive integer

## Homework Equations

## The Attempt at a Solution

since [tex]\binom{j}{r}=\binom{j}{j-r}[/tex]

we can rewrite the summation as

[tex](1-q)\sum\limits_{j=0}^\infty \binom{j}{j-r} p^r (1-p)^{j-r} q^j[/tex]

then i used a change of variables k=j-r and the summation became

[tex](1-q)\sum\limits_{k=-r}^\infty \binom{k+r}{k} p^r (1-p)^{k} q^{k+r}[/tex]

and now im stuck. i was hoping i could get the stuff inside the summation sign to look like the pdf of a negative binomial distribution

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