Bernoulli Distribution

Binomial Distribution with only a single trial

Probability Distribution Function

A random variable $X$ has a Bernoulli distribution, denoted by $X \sim B e r n o u l l i (p)$ if and only if it has the probability distribution function as

f (x) = p^{x} (1 - p)^{1 - x} for x = 0, 1

where $p = P (X = 1)$

E (X) = \sum_{x} x f (x) = p

\begin{aligned} V a r (X) & = E (X^{2}) - [E (X)]^{2} \\ = p - p^{2} \\ = p (1 - p) \end{aligned}