Bernoulli process
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In probability and statistics, a Bernoulli process is a discrete-time stochastic process consisting of a finite or infinite sequence of independent random variables X1, X2, X3,..., such that
- For each i, the value of Xi is either 0 or 1;
- For all values of i, the probability that Xi = 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 Xi 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 Xi 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.