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Bernoulli Distribution

The simplest discrete distribution, describing a single trial with exactly two possible outcomes, success or failure, occurring with probability $p$ and $1-p$ respectively.

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In Plain English

The Bernoulli distribution is the mathematical model of a single coin flip, or any yes/no, success/failure event: one trial, two possible outcomes, and a single number, $p$ , that fully describes how likely 'success' is. It's the smallest possible building block in probability theory, and nearly every other discrete distribution involving binary outcomes, the Binomial, Geometric, Negative Binomial, is built directly out of repeated or accumulated Bernoulli trials.

Definition

A discrete random variable $X \in {0, 1}$ follows a Bernoulli distribution with success probability $p \in [0, 1]$ if $P (X = 1) = p$ and $P (X = 0) = 1 - p$ , with mean $p$ and variance $p (1 - p)$ . It is the single-trial special case of the Binomial distribution ( $n = 1$ ), and a sequence of independent, identically distributed Bernoulli trials is the fundamental building block underlying the Binomial, Geometric, and Negative Binomial distributions.

Formula

P (X = x) = p^{x} (1 - p)^{1 - x}, x \in {0, 1}

Notation

p

—Probability of success (X=1)

Properties

The Bernoulli distribution is the single-trial special case of the Binomial distribution, every Binomial random variable is literally a sum of independent, identically distributed Bernoulli random variables, making the Bernoulli the atomic building block of the entire binary-outcome distribution family.
The Bernoulli distribution's variance, $p (1 - p)$ , is maximized at $p = 0.5$ (maximum uncertainty about the outcome) and approaches zero as $p$ approaches 0 or 1 (the outcome becomes near-certain), a shape that recurs throughout binary-outcome statistics.
As the response distribution in logistic regression, the Bernoulli distribution (technically a Binomial with $n = 1$ ) is one of the standard exponential-family distributions used as a Generalized Linear Model's likelihood for a binary outcome.

At a Glance

Support{0, 1}

Meanp

Variancep(1-p)

Special case ofthe Binomial distribution, with n = 1

Last updated August 9, 2026← Back to the Dictionary