---
tag: 0J97P5KZY0
kind: definition
name: Bernoulli distribution
msc: ["60E05"]
defines: 0386Y0Z7Y7
uses: [0WF4EJP91W, 0PHC9FE2V5]
sources:
  - url: https://en.wikipedia.org/wiki/Bernoulli_distribution
    locator: lead section
---

## Definition

Let $p \in [0, 1]$.
The Bernoulli distribution with parameter $p$ is the distribution on $\{0, 1\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(1) = p, \qquad p(0) = 1 - p.$$
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|0PHC9FE2V5]], with $1$ for success. It is the [[binomial distribution|0VRWZDWW6B]] 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 $p = 1/2$.