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.