Bernoulli distribution
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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 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 of information 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
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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.
Contents
Properties of the Bernoulli distributionEdit
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.
MeanEdit
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]}
VarianceEdit
SkewnessEdit
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 distributionsEdit
- 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]}
- 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 alsoEdit
- Bernoulli trials, random variables distributed according to a Bernoulli distribution
- Bernoulli process, a random process consisting of a sequence of independent Bernoulli trials
- Bernoulli sampling
- Binary entropy function
- Binomial distribution
- Binary decision diagram
ReferencesEdit
- ^ 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 readingEdit
- 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 linksEdit
Wikimedia Commons has media related to Bernoulli distribution. |
- Hazewinkel, Michiel, ed. (2001) [1994], "Binomial distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Weisstein, Eric W. "Bernoulli Distribution". MathWorld.
- Interactive graphic: Univariate Distribution Relationships