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 p and value 0 with failure probability q=1-p. So if X is a random variable with this distribution, we have:

\Pr(X=1) = 1 - \Pr(X=0) = 1 - q = p.\!

The probability mass function f of this distribution is

f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\ 1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.

This can also be expressed as

f(k;p) = p^k (1-p)^{1-k}\!.

The expected value of a Bernoulli random variable X is E\left(X\right)=p, and its variance is

\textrm{var}\left(X\right)=p\left(1-p\right).\,

The kurtosis goes to infinity for high and low values of p, but for p=1/2 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 X_1,\dots,X_n are independent, identically distributed (i.i.d.) random variables, all Bernoulli distributed with success probability p, then Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p) (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

• Bernoulli trial
• Bernoulli process
• Bernoulli sampling
• Sample size

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