Knowledge

❯

❯

Bernouli Distribution

Bernouli Distribution

Oct 27, 20251 min read

Math

A Bernoulli random variable takes two values: $0$ and $1$ , which are both specified by the below $p_{X}$ and functions

$X \sim Bernoulli (p) for p \in (0, 1)$

$p_{X} (x) = {p if x = 1 1 - p if x = 0$

$F_{X} (x) = ⎩ ⎨ ⎧ 0 if x < 0 1 - p if 0 \leq x < 1 1 if x \geq 1$

$M_{X} (t) =$

$E [X] = p$

$var (X) = p (1 - p)$

Graph View

$X\sim\text{Bernoulli}(p)\text{ for }p\in (0,1)$
$p_X(x)= \begin{cases} p~~~\qquad\text{if }x=1\\ 1-p\quad\text{if }x=0 \end{cases}$
$F_X(x) = \begin{cases} 0 ~~~~~~~\quad \text{if }x < 0\\ 1 - p\quad\text{if }0 \le x < 1\\ 1 ~~~\qquad \text{if }x \ge 1 \end{cases}$
$M_X(t) =$
$\mathbb{E}[X]=p$
$\text{var}(X)=p(1-p)$

Backlinks

Discrete Random Variables

Created with Quartz v4.5.2 © 2026

Personal Site
GitHub