# Single observation discrete distributions

## Bernoulli distribution

### Introduction

The outcome of a Bernoulli trial is either \(0\) or \(1\). We can describe it as:

\(P(1)=p\)

\(P(0)=1-p\)

With a single parameter \(p\).

### Moments of the Bernoulli distribution

The mean of a Bernoulli trial is \(E[X]=(1-p)(0)+(p)(1)=p\).

The variance of a Bernoulli trial is \(E[(X-\mu)^2]=(1-p)(0-\mu)^2+(p)(1-\mu)^2=(1-p)p^2+p(1-p)^2]=p(1-p)\).

## Discrete

### The categorical distribution

Bernoulli with three or more discrete possible outcomes.

### Degenerate distribution