Bernoulli process

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In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi are identical and independent. Prosaically, a Bernoulli process is repeated coin flipping, possibly with an unfair coin - but one whose unfairness is constant.

Every variable Xi in the sequence may be associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution.

The problem of determining the process, given only a limited sample of the Bernoulli trials, may be called the problem of checking if a coin is fair.

Contents

Definition

A Bernoulli process is a finite or infinite sequence of independent random variables X1X2X3, ..., 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.

Independence of the trials implies that the process is memoryless. Given that the probability p is known, past outcomes provide no information about future outcomes. (If p is unknown, however, the past informs about the future indirectly, through inferences about p.)

If the process is infinite, then from any point the future trials constitute a Bernoulli process identical to the whole process, the fresh-start property.

Interpretation

The two possible values of each Xi are often called "success" and "failure". Thus, when expressed as a number 0 or 1, the outcome may be called the number of successes on the ith "trial".

Two other common interpretations of the values are true or false and yes or no. Under any interpretation of the two values, the individual variables Xi may be called Bernoulli trials with parameter p.

In many applications time passes between trials, as the index i increases. In effect, the trials X1X2, ... Xi, ... happen at "points in time" 1, 2, ..., i, .... That passage of time and the associated notions of "past" and "future" are not necessary, however. Most generally, any Xi and Xj in the process are simply two from a set of random variables indexed by {1, 2, ..., n} or by {1, 2, 3, ...}, the finite and infinite cases.

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