Probability mass function
Three examples of Bernoulli distribution: and
and
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PMF |
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CDF | |||

Mean | |||

Median | |||

Mode | |||

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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" (or vice versa), respectively, and *p* would be the probability of the coin landing on heads or tails, respectively. 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.

## Properties

If is a random variable with this distribution, then:

The probability mass function of this distribution, over possible outcomes *k*, is

^{[2]}

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.

## 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

^{[2]}

## Variance

The variance of a Bernoulli distributed is

We first find

From this follows

^{[2]}

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

## Skewness

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

- If are independent, identically distributed (i.i.d.) random variables, all Bernoulli trials with success probability
*p*, then their sum is distributed according to a binomial distribution with parameters*n*and*p*:- (binomial distribution).
^{[2]}

- (binomial distribution).

- 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 also

- Bernoulli process, a random process consisting of a sequence of independent Bernoulli trials
- Bernoulli sampling
- Binary entropy function
- Binary decision diagram

## References

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

- 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 links

Wikimedia Commons has media related to Bernoulli distribution. |

- "Binomial distribution",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]. - Weisstein, Eric W. "Bernoulli Distribution".
*MathWorld*. - Interactive graphic: Univariate Distribution Relationships.