Bernoulli distribution

Definition

Let $p \in [0, 1]$. The Bernoulli distribution with parameter $p$ is the distribution on $\{0, 1\}$ with the probability mass function $$p(1) = p, \qquad p(0) = 1 - p.$$ Equivalently, $p(k) = p^k (1 - p)^{1-k}$ for $k \in \{0, 1\}$. One writes $X \sim \operatorname{Bernoulli}(p)$.

Remarks

The mean is $p$ and the variance is $p(1 - p)$. The indicator $\mathbf{1}_A$ of an event $A$ has the Bernoulli distribution with parameter $P(A)$. It is the distribution of the outcome of one Bernoulli trial, with $1$ for success. It is the binomial distribution with $n = 1$.

Explanations

L1 human

The Bernoulli distribution describes one yes-or-no experiment: $1$ means yes and $0$ means no. A fair coin, with $1$ for heads, has the Bernoulli distribution with $p = 1/2$.