# Bernoulli distribution

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In probability theory and statistics, the **Bernoulli distribution**, named after Swiss scientist Jacob Bernoulli, is the probability distribution of a random variable which takes value 1 with success probability and value 0 with failure probability . It can be used, for example, to represent the toss of a coin, where "1" is defined to mean "heads" and "0" is defined to mean "tails" (or vice versa).

## Properties

If is a random variable with this distribution, we have:

A classical example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability and tails with probability . The experiment is called fair if , indicating the origin of the terminology in betting (the bet is fair if both possible outcomes have the same probability).

The probability mass function of this distribution is

This can also be expressed as

The expected value of a Bernoulli random variable is , and its variance is

Bernoulli distribution is a special case of the Binomial distribution with .^{[1]}

The kurtosis goes to infinity for high and low values of , but for the Bernoulli distribution has 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.

## Related distributions

- If are independent, identically distributed (i.i.d.) random variables, all Bernoulli distributed with success probability
*p*, then

The Bernoulli distribution is simply .

- 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.
- The bernoulli distribution is a special case of binomial distribution.
- If
*Y*~ Bernoulli(0.5), then (2*Y*-1) has a Rademacher distribution.

## See also

## Notes

- ↑ McCullagh and Nelder (1989), Section 4.2.2.

## References

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- Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9

## External links

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- Weisstein, Eric W., "Bernoulli Distribution",
*MathWorld*. - Interactive graphic: Univariate Distribution Relationships

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