In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability that is, the probability distribution of any single experiment that asks a yes–no question; the question results in a boolean-valued outcome, 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.
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
The Bernoulli distribution is a special case of the binomial distribution with
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 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
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 central moment of order is given by
The first six central moments are
The higher contral moments can be espressed more compactly in terms of and
The first six cumulants are
- 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:
- 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.
- James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
- 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.
- 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.
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