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:

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 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 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.