---
title: "Bernoulli distribution"
type: concept
related: [Probability Mass Function, Variance, Maximum Likelihood Parameter Learning, Bernoulli Random Variable, Expectation]
source: https://www.jemoka.com/posts/kbhbernoulli_random_variable/
confidence: high
status: active
---

Consider a case where there&rsquo;s only a single binary outcome:
&ldquo;success&rdquo;, with probability \(p\) &ldquo;failure&rdquo;, 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&rsquo;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&rsquo;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
