In mathematics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability p and value 0 with failure probability q = 1 − p. The probability mass function f of this distribution is

<math> f(x) = \left\{\begin{matrix} p & \mbox {if }x=1, \\

q & \mbox {if }x=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.<math>

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

pq = p(1 − p).

The Bernoulli distribution is a member of the exponential family.

See also Bernoulli trial, Bernoulli process.

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