# Help simplifying this summation

## Homework Statement

$$\sum\limits_{j=0}^\infty \binom{j}{r} p^r (1-p)^{j-r} (1-q) q^j$$

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

## The Attempt at a Solution

since $$\binom{j}{r}=\binom{j}{j-r}$$

we can rewrite the summation as

$$(1-q)\sum\limits_{j=0}^\infty \binom{j}{j-r} p^r (1-p)^{j-r} q^j$$

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

$$(1-q)\sum\limits_{k=-r}^\infty \binom{k+r}{k} p^r (1-p)^{k} q^{k+r}$$

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

Last edited:

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vela
Staff Emeritus
Homework Helper
Do you have any identities that might be relevant to evaluating the summation?

Also, the notation where q = 1-p is fairly common. Does that apply here or are p and q just two unrelated variables?

tiny-tim
Homework Helper
hi robertdeniro!

since r is a constant, you can take all the r stuff outside the ∑

(but I don't think it converges)

Do you have any identities that might be relevant to evaluating the summation?

Also, the notation where q = 1-p is fairly common. Does that apply here or are p and q just two unrelated variables?
nope, here p and q are not related

EDIT: Guys, please see me attempt at the solution and let me know what you think

vela
Staff Emeritus
Homework Helper
I don't see how that helps, but I might just be missing something.

Are you familiar with the generating functions for the binomial coefficients?

I don't see how that helps, but I might just be missing something.

Are you familiar with the generating functions for the binomial coefficients?
yes but i dont see how that would help

EDIT: nevermind, thanks for that tip! i think i got it

vela
Staff Emeritus