9 Bernoulli random variables

9.1 Learning objectives

• Define and understand Bernoulli random variables.

9.2 Bernoulli random variable

9.2.1 Definition

A single sample from a any random variable that produces a dichotomous outcome is formally defined as a Bernoulli trial. To be a Bernoulli trial, a few conditions must be met:

1. Each trial yields one of the two outcomes usually called success ($S$) and failure ($F$).

2. For each trial, the probability of success $P(S)$ is the same and is denoted by $p = P(S)$. The probability of failure is then $q = P(F) = 1 - P(S)$ for each trial.

3. Trials are independent. The probability of success in a trial does not change given any amount of information about the outcomes of other trials.

Bernoulli random variables are completely defined by the single parameter $p = P(S)$. If $X$ is a Bernoulli random variable then we would write:

$X \sim Bernoulli(p)$

9.2.2 Probability distribution

The probability distribution of a Bernoulli random variable is very simple. We simply need to indicate two probabilities as follows.