Bernoulli distribution

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]| kurtosis =[\frac]| entropy =[-q\ln(q)-p\ln(p)\,]| mgf =[q+pe^t\,]| char =[q+pe^\,]| }}

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) = p.\!]

The probability mass function f of this distribution is

[ f(k;p) = \left\ p & \mbox k=1, \\1-p & \mbox k=0, \\0 & \mbox \end\right.]

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

[\textrm\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 random variables, all Bernoulli distributed with success probability p, then [Y = \sum_^n X_k \sim \mathrm(n,p)] (binomial distribution).

See also

• Bernoulli trial
• Bernoulli process

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