# What is the distribution of the binomial distribution parameter N given sample k and p?

Say $k \sim Bin(N, p)$. What is the distribution of $N$, given fixed $p$ and $k$? Looks like Poisson but starting at k instead of zero (???)

Thanks.

EDIT: Application: I have some real number of birds ($N$), each is seen with probability $p$. I see $k$ birds and want to estimate the real number of birds ($N$).

• $N$ has no distribution unless you assume one or unless you have some kind of stopping rule for a sequential experiment. What's your situation? – whuber Aug 23 '11 at 19:27
• My situation is as I described - I have k which I know is a draw from some Bin(N, p), I also know p and I want to know distribution of N... – Curious Aug 23 '11 at 21:28
• That's not sufficient information to answer the question unambiguously. If that's all you know, there are two classes of answer. One is that $N$ does not have a distribution (but you could at least compute a confidence interval for $N$). The other is that if you assume some prior distribution for $N$, you can use Bayes' Theorem to update it based on $k$ and $p$. – whuber Aug 23 '11 at 21:31
• Well, I'm afraid that's all I know now, I also don't have any prior, moreover I don't much believe in priors yet :-) – Curious Aug 23 '11 at 21:40
• Say that I would use as much uninformative prior as possible... like uniform distrubution from 0 to some very high number or something like that... how the distribution of N would look like then? – Curious Aug 23 '11 at 21:44

• OK, so the answer would then be N ~ k + NBin(k, 1 - p), is that correct? – Curious Aug 23 '11 at 21:18