Bernoulli distribution

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Probability mass function
Cumulative distribution function
Parameters (real)
Probability mass function (pmf)
Cumulative distribution function (cdf)
Median N/A
Excess kurtosis
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . So if X is a random variable with this distribution, we have:

The probability mass function f of this distribution is

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

The kurtosis goes to infinity for high and low values of p, but for the Bernoulli distribution has a lower kurtosis than any other probability distribution, namely -2.

The Bernoulli distribution is a member of the exponential family.

Related distributions

  • If are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then (binomial distribution).
  • 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.

See also

ar:توزيع برنولي de:Bernoulli-Verteilung it:Variabile casuale bernoulliana he:התפלגות ברנולי nl:Bernoulli-verdeling nov:Distributione de Bernoulli fi:Bernoullin jakauma uk:Розподіл Бернуллі