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 probabilityp 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", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). 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.
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Propertiesedit
If is a random variable with a Bernoulli distribution, then:
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, namely −2, than any other probability distribution.
With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside .
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
Higher moments and cumulantsedit
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
^Uspensky, James Victor (1937). Introduction to Mathematical Probability. New York: McGraw-Hill. p. 45. OCLC 996937.
^Dekking, Frederik; Kraaikamp, Cornelis; Lopuhaä, Hendrik; Meester, Ludolf (9 October 2010). A Modern Introduction to Probability and Statistics (1 ed.). Springer London. pp. 43–48. ISBN 9781849969529.
^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.
^Orloff, Jeremy; Bloom, Jonathan. "Conjugate priors: Beta and normal" (PDF). math.mit.edu. Retrieved October 20, 2023.
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
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