In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable. 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

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.

Properties

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

The kurtosis the two-point distributions including the Bernoulli distribution have a lower excess kurtosis, namely −2, than any other probability distribution.

form an exponential family.

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

Mean

The expected value is

we find

Variance

The variance is

We first find

From this follows

.

Skewness

The skewness. Thus we get

Higher moments and cumulants

.

Entropy and Fisher's Information

Entropy

is defined as:

, where one outcome is certain.

Fisher's Information

Fisher information is given by:

Proof:

• The Likelihood Function.

• The Log-Likelihood Function

• Fisher information

.

Related distributions

• are independent, identically distributed (i.i.d.) random variables, all Bernoulli trials with success probabilityp, then their sum is distributed according to a binomial distribution with parameters n and p:

  1. (binomial distribution).

  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.

• has a Rademacher distribution.

See also

• Bernoulli process, a random process consisting of a sequence of independent Bernoulli trials

• Bernoulli sampling

• Binary entropy function

• Binary decision diagram

References

Author's mention

External links

• .

• Interactive graphic: Univariate Distribution Relationships.