# Bernoulli distribution

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In probability theory and statistics, the **Bernoulli distribution**, named after Swiss scientist Jacob Bernoulli,^{[1]} is the probability distribution of a random variable which takes the value 1 with success probability of and the value 0 with failure probability of . It can be used to represent a coin toss where 1 and 0 would represent "head" and "tail" (or vice versa), respectively. In particular, unfair coins would have .

The Bernoulli distribution is a special case of the **two-point distribution**, for which the two possible outcomes need not be 0 and 1. It is also a special case of the binomial distribution; the Bernoulli distribution is a binomial distribution where n=1.

## Properties of the Bernoulli Distribution

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

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

This can also be expressed as

The Bernoulli distribution is a special case of the binomial distribution with .^{[2]}

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

## Variance

The variance of a Bernoulli distributed is

We first find

From this follows

## 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

## 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.
- If
*Y*~ Bernoulli(0.5), then (2*Y*-1) has a Rademacher distribution.

## See also

## Notes

- ↑ James Victor Uspensky:
*Introduction to Mathematical Probability*, McGraw-Hill, New York 1937, page 45 - ↑ McCullagh and Nelder (1989), Section 4.2.2.

## References

- McCullagh, Peter; Nelder, John (1989).
*Generalized Linear Models, Second Edition*. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5. - Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9

## External links

Wikimedia Commons has media related to .Bernoulli distribution |

- Hazewinkel, Michiel, ed. (2001), "Binomial distribution",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Weisstein, Eric W. "Bernoulli Distribution".
*MathWorld*.

- Interactive graphic: Univariate Distribution Relationships