with the binomial coefficients
The distribution has the properties
|mean:||E(X) = np,|
|curtosis:||c = (1-6pq)/(npq) +3,|
which are determined by the single parameter p. If in a sample of n events k have the property A, then the maximum likelihood estimator of the parameter p is given by
The variance of the estimator of p is
for which an unbiased estimator is
Note that the probability of obtaining k events out of n for a given p should not be estimated by comparing the difference of P and p against s2(P), but from a Poisson distribution with mean pn, particularly if P is close to 0 or 1.