# 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.