Bernoulli distribution
Probability mass function
Funzione di densità di una variabile casuale normale

Three examples of Bernoulli distribution:


Ex. kurtosis
Fisher information

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,[1] is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability . Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have

The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1.


If is a random variable with a Bernoulli distribution, then:

The probability mass function 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 [3]

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

The Bernoulli distributions for form an exponential family.

The maximum likelihood estimator of based on a random sample is the sample mean.


The expected value of a Bernoulli random variable is

This is due to the fact that for a Bernoulli distributed random variable with and we find



The variance of a Bernoulli distributed is

We first find

From this follows


With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside .


The skewness is . When we take the standardized Bernoulli distributed random variable we find that this random variable attains with probability and attains with probability . Thus we get

Higher moments and cumulants

The raw moments are all equal due to the fact that and .

The central moment of order is given by

The first six central moments are

The higher central moments can be expressed more compactly in terms of and

The first six cumulants are

Related distributions

The Bernoulli distribution is simply , also written as

See also


  1. ^ Uspensky, James Victor (1937). Introduction to Mathematical Probability. New York: McGraw-Hill. p. 45. OCLC 996937.
  2. ^ a b c d Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.
  3. ^ McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. Section 4.2.2. ISBN 0-412-31760-5.

Further reading