Bernoulli distribution

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.$$

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  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.