# Estimation of the number of trials n made after having observed s successes with probability p

#### The problem

Consider the situation where we have observed s successes and know the probability of success p, but would like to know how many trials were actually done to have observed those successes. We wish to estimate a value that is fixed, so we require a distribution that represents our uncertainty about what the true value is. There are two possible situations: we either know that the trials stopped on the sth success or we do not.

#### Results

If we know that the trials stopped on the sth success, we can model our uncertainty about the true value of n as:

n = NegBinomial(p,s)

If, on the other hand, we do not know that the last trial was a success (though it could have been), then our uncertainty about n is modeled as:

n = NegBinomial(p, s+1) - 1

Both of these formulae result from a Bayesian analysis with Uniform priors for n.

#### Derivations

Let x be the number of trials that were needed to obtain the sth success. We will use a uniform prior for x, i.e. p(x) = c, and, from the binomial distribution, the likelihood function is the probability that at the (x-1)th trial there had been (s-1) successes and then the xth trial was a success, which is just the Negative Binomial probability mass function:

Since we are using a uniform prior (assuming no prior knowledge), and the equation for l(X|x) comes directly from a distribution (so it must sum to unity) we can dispense with the formality of normalizing the posterior distribution to one, and observe:

i.e. that x = NegBinomial(p,s).

In the second case, we do not know that the last trial was a success, only that in however many trials were completed, there were just s successes. We have the same Uniform prior for the number of trials, but our likelihood function is just the binomial probability mass function, i.e.:

Since this does not have the form of a probability mass function of a known distribution, we need to complete the Bayesian analysis, so:

Look at the denominator, and substituting j = i+1 gives:

since

is the probability mass function for the NegBinomial(s+1,p) distribution for j and therefore sums to 1. The posterior distribution then reduces to:

For x = y - 1:

i.e. y = NegBinomial(p,s+1), and therefore x = NegBinomial(p,s+1) -1 distribution.

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