Binomial Distribution

  A given experiment may yield the event A or the event (not A) with the probabilities P(A)=p and , respectively. If the experiment is repeated n times and X is the number of times A is obtained, then the probability of X taking exactly a value k is given by

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 s²(P), but from a Poisson distribution with mean pn, particularly if P is close to 0 or 1.