Bernoulli trial

We now concentrate on a single event in a random situation. Either the event happens, which we call a "success" or it does not happen, a "failure". These words are simply intended to provide a general terminology that can be applied to any kind of event and there should be no positive connotation to a "success". When dealing with the survival of a cancer patient two years after chemotherapy, we might denote either death or survival as "success".

Examples of "successes"

A random experiment such as this with only two possible outcomes is called a Bernoulli trial.

Bernoulli distribution

It is useful to define a numerical variable based on one such event. This has the value one when the event happens and zero when it does not happen.

Definition

If there is an event \(A\) for which \(P(A) = \pi\), and a random variable \(X\) is defined by

\[ X = \begin {cases} 1 & \quad \text{if } A \text{ is a success}\\[0.5em] 0 & \quad \text{if } A \text{ is a failure} \end {cases} \]

then \(X\) has a Bernoulli distribution with parameter \(\pi\),

\[ X \;\; \sim \; \; \BernoulliDistn(\pi) \]

From this definition, \(X\) is a discrete random variable with the following probability function:

\[ p(x) = \begin {cases} \pi & \quad \text{if } x = 1\\[0.5em] (1 - \pi) & \quad \text{if } x = 0\\[0.5em] 0 & \quad \text{otherwise} \end {cases} \]

Sequence of Bernoulli trials

Although a single Bernoulli trial is too simple to be of much practical importance, situations often arise that can be treated as sequences of independent Bernoulli trials, all of which have the same probability of success. Later sections of this chapter describe distributions related to a sequence of Bernoulli trials.

Examples