with the binomial coefficients

The distribution has the properties

mean: | E(X) = np, |

variance: | , |

skewness: | , |

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 *s*^{2}(*P*), but from a Poisson distribution with mean *pn*, particularly if *P* is close to 0 or 1.

Rudolf K. Bock, 7 April 1998