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(x,p) =   

p, if x = 1   

1-p, if x = 0   

0, otherwise

The expected value of a Bernoulli random variable X is E(X)=p, and its variance is var(X)=p(1-p).

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 -5.

The Bernoulli distribution is a member of the exponential family.

support x = 0,…,n