2.2b.2 Bernoulli Distribution

Bernoulli Distribution

A discrete random variable $X\in \{0,1\}$ follows the Bernoulli distribution $X\sim \text{Bernoulli}(p)$ if it represents a single trial with binary outcomes. Formally, the Bernoulli distribution of a random variable is given by the probability mass function,
$p_X(x)=\{\begin{array}{ll}p& \text{if }x=1,\\ 1-p& \text{if }x=0,\end{array}$
where $p$ is the probability of a successful outcome.

Explanation.

The Bernoulli distribution is used for probabilities where the event either happens, or does not happen. Following this, the random variable can only take two values (1 or 0).

Properties

(Mean) $\mathbb{E}[X\sim \text{Bernoulli}(p)]=p.$

(Variance) $\text{Var}[X\sim \text{Bernoulli}(p)]=p(1-p)$.