Bernoulli process - Definition and Overview

In probability and statistics, a Bernoulli process is a discrete-time stochastic process consisting of finite or infinite sequence of independent random variables X₁, X₂, X₃,..., such that

• For each i, the value of X_i is either 0 or 1;
• For all values of i, the probability that X_i = 1 is the same number p.

In other words, a Bernoulli process is a sequence of independent identically distributed Bernoulli trials. The two possible values of each X_i are often called "success" and "failure", so that, when expressed as a number, 0 or 1, the value is said to be the number of successes on the ith "trial". The individual success/failure variables X_i are also called Bernoulli trials.

Random variables associated with the Bernoulli process include

• The number of successes in the first n trials; this has a binomial distribution;
• The number of trials needed to get r successes; this has a negative binomial distribution.
• The number of trials needed to get one success; this has a geometric distribution, which is a special case of the negative binomial distribution.