Consider a case where there’s only a single binary outcome:

- “success”, with probability \(p\)
- “failure”, with probability \(1-p\)

## constituents

\begin{equation} X \sim Bern(p) \end{equation}

## requirements

the probability mass function:

\begin{equation} P(X=k) = \begin{cases} p,\ if\ k=1\\ 1-p,\ if\ k=0\\ \end{cases} \end{equation}

This is sadly not Differentiable, which is sad for Maximum Likelihood Parameter Learning. Therefore, we write:

\begin{equation} P(X=k) = p^{k} (1-p)^{1-k} \end{equation}

Which emulates the behavior of your function at \(0\) and \(1\) and we kinda don’t care any other place.

We can use it

## additional information

### properties of Bernoulli distribution

**expected value**: \(p\)**variance**: \(p(1-p)\)

### Bernoulli as indicator

If there’s a series of event whose probability you are given, you can use a Bernoulli to model each one and add/subtract

### MLE for Bernouli

\begin{equation} p_{MLE} = \frac{m}{n} \end{equation}

\(m\) is the number of events