Bernoulli distribution


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

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

Three examples of Bernoulli distribution:



Ex. kurtosis
Fisher information

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 this 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 cumulantsEdit

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 distributionsEdit

The Bernoulli distribution is simply  , also written as  
  • The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.
  • The Beta distribution is the conjugate prior of the Bernoulli distribution.
  • The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success.
  • If  , then   has a Rademacher distribution.

See alsoEdit


  1. ^ James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
  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 readingEdit

  • Johnson, N. L.; Kotz, S.; Kemp, A. (1993). Univariate Discrete Distributions (2nd ed.). Wiley. ISBN 0-471-54897-9.
  • Peatman, John G. (1963). Introduction to Applied Statistics. New York: Harper & Row. pp. 162–171.

External linksEdit