Bernoulli distribution

For $\theta\in[0,1]$ , the Bernoulli distribution $\operatorname{Bernoulli}(\theta)$ is the distribution on $\{0,1\}$ with probability mass function

p(1)=\theta,\qquad p(0)=1-\theta.

Interpretation

The value one denotes success and zero denotes failure. The indicator of an event of probability $\theta$ has this distribution, including the deterministic cases $\theta=0$ and $\theta=1$ .

Moments

If $X$ has this law, its expectation is $\theta$ and its variance is $\theta(1-\theta)$ .