The binomial distribution with $n$ trials, $k$ successes and success probability $p$ is given by

$$P(k;n,p) = \binom{n}{k} p^k (1-p)^{(n-k)}, \quad k \in \{0,...,n\}$$

Suppose that we observe $k$ successes and know $p$ but we do not know $n$. Observe that now $k$ and $p$ are fixed whereas $n$ is stochastic. So if $k=6$ and $p=0.4$,

$$P(k=6; n ,p=0.4) = \binom{n}{6} 0.4^6 (0.6)^{(n-6)}, \quad n \in \{6,...,\infty\}.$$ This is however (remark by @Xiaomi) not a valid probability function as it does not sum to one over its suppoer. Is there a probability mass function for $n$? What is a useful (unbiased, consistent) estimator for its parameter $n$?

  • $\begingroup$ I think you are misunderstanding the notation. '$|$' here doesn't mean anything conditional, it means variables that are given. You can't use Bayes's theorem. It's not really clear of what you are asking either. One could try to guess, that you are talking about some sort of extension of binomial distribution to all natural numbers. But that's speculations $\endgroup$ – Jakobian Oct 4 '18 at 14:58
  • $\begingroup$ @Jakobian I completely revised the question to make it clearer what I am asking. $\endgroup$ – tomka Oct 4 '18 at 15:29

As noted in Xiaomi's answer, the probability distribution

$P(k=6; n ,p=0.4) = \binom{n}{6} 0.4^6 (0.6)^{(n-6)}, \quad n \in \{6,...,\infty\}.$

fails. The problem is that it assumes the six successes occur randomly among the $n$ occurrences, but this is not true. To achieve $n$ as an outcome the sixth success must occur exactly on attempt $n$. Only the first five successes occur randomly, and they are restricted to the first $n-1$ attempts (but no need for the fifth success to occur exactly at attempt $n-1$. The correct probability distribution with these characteristics is

$$P(k=6; n ,p=0.4) = \binom{n-1}{5} 0.4^5 (0.6)^{((n-1)-5)}\color{blue}{(0.4)}, \quad n \in \{6,...,\infty\}.$$

where the blue factor forces a success on trial $n$ and the rest of the expression accounts for the proper random occurrence of the other five successes. This simplifies to

$$P(k=6; n ,p=0.4) = \binom{n-1}{5} 0.4^6 (0.6)^{(n-6)}, \quad n \in \{6,...,\infty\}.$$

which now does normalize properly and should give consistent statistical estimates.

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  • $\begingroup$ Great! How would you arrive at an estimator though? It seems that the expectation of $n$ is the solution of an infinite sum starting at $k$, $E(n) = \sum_{n=k}^{\infty} \binom{n}{k} p^k (1-p)^{(n-k)}$. I am not even sure if this sum converges. $\endgroup$ – tomka Oct 4 '18 at 19:53
  • $\begingroup$ Jusr sum $kP(n=k)$. The series converges by the ratio test with ratio = $1−p=0.6$ in this case. The mean will be what you expect, namely $r/p$ where you demand $r$ successes, try it. The variance is derived by summing $k^2P(n=k)$ and subtracting the mean squared, which gives $r(1−p)/p$. $\endgroup$ – Oscar Lanzi Oct 4 '18 at 21:50
  • $\begingroup$ "To achieve $n$ as an outcome the sixth success must occur exactly on attempt $n$". Why ? what about the sequence 6th success at $n-1$ plus fail at $n$, etc. ? $\endgroup$ – G Cab Jan 31 '19 at 1:34

First of all, what you've stated is not the distribution function of $n$. It's the distribution function of $X$ given parameters $n,p$. You cannot simply interchange $n$ and $k$. If it was the PMF of $n$, it would sum to $1$ over all values of $n$, and that clearly doesn't. To answer your question...

In the (very unrealistic) situation where we have a Binomial random variable $X$, the number of successes out of $n$ trials, and we know $p$ in advance, we can estimate $n$ as simply as

$$\hat{n} = \frac{X}{p}$$

The basic idea here being that we observe $X$ successes, and so to get back to $n$ we need to re-scale by $1/p$. However this entire thought process is a bit non-sensical, as a Binomial random variable is characterised as being a number of successes out of some fixed and known number of trials $n$.

An interesting question is whether this estimator is consistent. Clearly it is unbiased, since

$$E[X/p] = np/p = n$$

But for the variance, we have

$$Var(\hat{n}) = Var(X/p) = Var(X)/p^2 = np(1-p)/p^2$$

So our estimator is clearly not consistent.

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  • $\begingroup$ Thanks. In an earlier version of my question I asked what is the distribution of $n$. Maybe this version was not so bad after all. However, ultimately I am interested in an estimator for $n$; however I believe a good estimator should be consistent. $\endgroup$ – tomka Oct 4 '18 at 16:06
  • $\begingroup$ I amended my question a bit following your answer. $\endgroup$ – tomka Oct 4 '18 at 16:09
  • $\begingroup$ I found the way to confirm your results for the variance. For the expected value there is instead an additional term. $\endgroup$ – G Cab Jan 31 '19 at 23:56

The binomial distribution is the probability of having $s$ successes in $n$ trials, given that the probability of success in each trial is $p$, and the outcomes of the trials are i.i.d. (Bernoulli Trials) .

The parameter $n$ is given, so wrt this the distribution is a conditional probability and we can write $$ P\left( {s\,\left| {\,n} \right.} \right) = \left( \matrix{ n \cr s \cr} \right)p^{\,s} q^{\,n - s} = {{P\left( {s \wedge n} \right)} \over {P(n)}} $$

We want to determine the complementary conditional probability $$ P\left( {n\,\left| {\,s} \right.} \right) = {{P\left( {s \wedge n} \right)} \over {P(s)}} $$ which is a totally licit question, provided that we know $P(n)$.

Assume that $n$ is uniformly distributed over the interval $[0,N]$.
Thus $P(n)= 1/(N+1)$, and we get $$ P\left( {s \wedge n} \right) = {{\left[ {0 \le n \le N} \right]} \over {N + 1}}\binom{n}{s}p^{\,s} q^{\,n - s} $$ where $[P]$ denotes the Iverson bracket

Note that the sum of the bivariate distribution $$ \eqalign{ & \sum\limits_{0\, \le \,n\,\left( { \le \,N} \right)} {\sum\limits_{0\, \le \,s\,\left( { \le \,n} \right)} {P\left( {s \wedge n} \right)} } = {1 \over {N + 1}}\sum\limits_{0\, \le \,n\,\left( { \le \,N} \right)} {\left[ {0 \le n \le N} \right]\sum\limits_{0\, \le \,s\,\left( { \le \,n} \right)} { \binom{n}{s} p^{\,s} q^{\,n - s} } } = \cr & = {1 \over {N + 1}}\sum\limits_{0\, \le \,n\,\left( { \le \,N} \right)} {\left[ {0 \le n \le N} \right]} = 1 \cr} $$ correctly checks to be $1$.

Then the marginal distribution in $s$ will be $$ P(s) = \sum\limits_{0\, \le \,n\,\left( { \le \,N} \right)} {P\left( {s \wedge n} \right)} = {{p^{\,s} q^{\, - s} } \over {N + 1}}\sum\limits_{0\, \le \,n\, \le \,N} {\binom{n}{s}q^{\,n} } $$ and we reach to $$ P\left( {n\,\left| {\,s} \right.} \right) = {{P\left( {s \wedge n} \right)} \over {P(s)}} = \left[ {0 \le n \le N} \right]{{\binom{n}{s}q^{\,n} } \over {\sum\limits_{0\, \le \,n\, \le \,N} {\binom{n}{s}q^{\,n} } }} $$

In the limit for $N \to \infty$ the expression above converges to $$ \bbox[lightyellow] { P\left( {n\,\left| {\,s} \right.} \right) = \binom{n}{s} \, q^{\,n - s} p^{\,s + 1} }$$

The expected value and the variance for $n$ result to be: $$ \bbox[lightyellow] { \eqalign{ & E\left( {n\left| {\,s} \right.} \right) = \sum\limits_{0\, \le \,n\,} {n\binom{n}{s}q^{\,n - s} p^{\,s + 1} } = {{1 - p} \over p} + {1 \over p}s \cr & \sigma ^{\,2} = \sum\limits_{0\, \le \,n\,} {\left( {n - {{1 - p + s} \over p}} \right)^{\,2} \binom{n}{s}q^{\,n - s} p^{\,s + 1} } = {{\left( {1 - p} \right)\left( {s + 1} \right)} \over {p^{\,2} }} \cr} }$$

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  • $\begingroup$ @tomka since the problem is interesting, I recasted my answer to render it more rigorous $\endgroup$ – G Cab Jan 31 '19 at 23:53

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