# 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.
- 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.