linguistic:translation, information:extraction
Properties

If X {\displaystyle X} is a random variable with a Bernoulli distribution, then:

The probability mass function f {\displaystyle f} of this distribution, over possible outcomes k, is

This can also be expressed as

or as

The Bernoulli distribution is a special case of the binomial distribution with n = 1. {\displaystyle n=1.}

The kurtosis goes to infinity for high and low values of p, {\displaystyle p,} but for p = 1 / 2 {\displaystyle p=1/2} the two-point distributions including the Bernoulli distribution have a lower excess kurtosis, namely −2, than any other probability distribution.

The Bernoulli distributions for 0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} form an exponential family.

The maximum likelihood estimator of p {\displaystyle p} based on a random sample is the sample mean.