# Bernoulli Distribution

All cases in which manifestations have exactly two characteristics follow a Bernoulli distribution.

Typical examples are coin flip or medical treatment, which works or not.

Let X be a random variable with $x \in {0,1}$ and a probability function of

$f(1)=P(X=1)=\pi$

$f(0)=P(X=0)=1-\pi$

Written in a compact way:

$f(x)=\left\{\begin{array}{ll} \pi^{x}(1-\pi)^{1-x} & x\in {0,1} \\ 0 & otherwise\end{array}\right. .$

The expected value is defined through:

$\theta=\pi=P(X=1)=E(X)=0*P(X=0)+1*P(X=1)$

In case of having a series of Bernoulli events, we are having a Binomial distribution. For example, the out come of a repeated coin flip follows a Binomial distribution.