Help simplifying this summation

  • #1

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
 
Last edited:

Answers and Replies

  • #2
vela
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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?
 
  • #3
tiny-tim
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hi robertdeniro! :smile:

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

(but I don't think it converges)
 
  • #4
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
 
  • #5
vela
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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?
 
  • #6
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
 
  • #7
vela
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What generating functions do you know? At least one I've seen applies directly to the problem.
 

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