Bernoulli Distribution¶

Definition¶

Let $0\le p\le1$ $1$ . A random variable $X$ has a Bernoulli distribution with parameter $p$ , written $X\sim\operatorname{Bern}(p)$ $Bern (p)$ , if

\mathbb{P}(X=1)=p,\qquad \mathbb{P}(X=0)=1-p.

We write $q=1-p$ $p$ . The endpoints $p=0$ and $p=1$ give degenerate distributions: $X$ is then constant, always $0$ or always $1$ .

Interpretation¶

A Bernoulli variable models a single trial with two possible results, success coded as $1$ and failure as $0$ . The one number $p=\mathbb{P}(X=1)$ determines the whole distribution.

Two probability bars for a Bernoulli(0.3) variable: 0.7 at the value 0 and 0.3 at the value 1.

The figure shows $X\sim\operatorname{Bern}(0.3)$ $Bern (0.3)$ : a bar of height $0.7$ at the value $0$ and a bar of height $0.3$ at the value $1$ ; the two bars add up to $1$ . The formulas in the box are the Key Results of this page.

Coding success as $1$ makes $X$ the indicator of the event $\{X=1\}$ $1}$ ; conversely, the indicator of any event with probability $p$ is $\operatorname{Bern}(p)$ . The label "success" is arbitrary: $1-X$ is again Bernoulli, with parameter $1-p$ $p$ . The coding pays off when trials are repeated: for independent trials $X_1,\ldots,X_n$ , all $\operatorname{Bern}(p)$ , the sum $X_1+\cdots+X_n$ counts the successes and has the binomial distribution.

Key Results¶

$\mathbb{E}X=p$ and $\operatorname{Var}(X)=p(1-p)$ $p)$ .
MGF $M_X(t)=1-p+p\mathrm{e}^t$ for every real $t$ .

Examples¶

A fair coin indicator has mean $1/2$ and variance $1/4$ .
A defect indicator with $p=0.02$ has $\mathbb{P}(X=0)=0.98$ and variance $0.0196$ .

Further Properties¶

An indicator of an event with probability $p$ has this distribution; sums of independent Bernoulli variables with the same success probability $p$ produce binomial counts.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3
Nicky van Foreest · Probability Distributions · Refresher

Aliases: Bern(p), Bernoulli random variable, Bernoulli variable, Bernoulli r.v. Keywords: success indicator

Last updated: 20 October 2026 03:02