Bernoulli process

Bernoulli process

In probability and statistics, a Bernoulli process or a Bernoulli experiment is a 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.

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.