Bernoulli Distribution

Compact study note.

Summary

Bernoulli distribution models a success/failure trial coded with $1$ for success and $0$ for failure.^[1]

Prerequisites

• Discrete Random Variable

Definition

$X\sim \mathrm{Bernoulli}(p)$ when $P(X=1)=p$ and $P(X=0)=1-p$ .

Notation and Assumptions

Use single trial only. For multiple independent Bernoulli trials, use binomial model.

Parameters

$p$ is the success probability with $0\le p\le 1$ .

Support

$\{0,1\}$ .

PMF or PDF

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

CDF

$F_X(x)=0$ for $x<0$ , $1-p$ for $0\le x<1$ , and $1$ for $x\ge 1$ .

Moments

$E[X]=p$ , $\mathrm{Var}(X)=p(1-p)$ , and $M_X(t)=1-p+p{e}^t$ .

Essential Result

One Bernoulli variable is the building block for binomial, geometric, and negative binomial distributions.

Small Example

For biased coin with $P(H)=0.7$ , let $X=1$ for heads. Then

Common Mistakes

• Using two coin flips as Bernoulli variable when $X$ counts heads.
• Writing the MGF as probability generating function without $\exp (t)$ .

Connections

• Binomial Distribution
• Geometric Distribution
• Expectation from an MGF

References