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In probability theory and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution.
- For each , the value of is either 0 or 1;
- For all values of , the probability that is the same number .
Independence of the trials implies that the process is memoryless. Given that the probability is known, past outcomes provide no information about future outcomes. (If is unknown, however, the past informs about the future indirectly, through inferences about .)
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.
The two possible values of each 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 th "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 may be called Bernoulli trials with parameter .
In many applications time passes between trials, as the index i increases. In effect, the trials happen at "points in time" . That passage of time and the associated notions of "past" and "future" are not necessary, however. Most generally, any and in the process are simply two from a set of random variables indexed by or by , the finite and infinite cases.
Several random variables and probability distributions beside the Bernoullis may be derived from the Bernoulli process:
- The number of successes in the first n trials, which has a binomial distribution B(n, p)
- The number of failures needed to get r successes, which has a negative binomial distribution NB(r, p)
- The number of failures needed to get one success, which has a geometric distribution NB(1, p), a special case of the negative binomial distribution
Using the example of the Tech card in a Daily Kit Box † which can be either Advanced Tech or Mid-Tech , the Bernoulli process can be formalized in the language of probability spaces as a random sequence of independent realisations of a random variable that can take values of Advanced or Mid-Tech. The state space for an individual value is denoted by .
Specifically, one considers the countably infinite direct product of copies of . It is common to examine either the one-sided set or the two-sided set .
If the chances of revealing Advanced Tech or Mid-Tech are given by the probabilities , then one can define a natural measure on the product space, given by (or by for the two-sided process). Given a specific sequence of revealed cards at times , the probability of observing this particular sequence is given by
where is the number of times that appears in the sequence, and is the number of times that appears in the sequence. There are several different kinds of notations for the above; a common one is to write
where each is a binary-valued random variable with
- if or
- if .
Note that the probability of any specific, infinitely long sequence of revealed cards is exactly zero; this is because
- for any .
Law of large numbers
- Main article: Law of large numbers
Let us assume the canonical process with represented by and represented by . The law of large numbers states that, on the average of the sequence, i. e., , will approach the expected value almost surely, that is, the events which do not satisfy this limit have zero probability. The expected value of revealing Advanced Tech, assumed to be represented by 1, is given by . In fact, one has
for any given random variable out of the infinite sequence of Bernoulli trials that compose the Bernoulli process.
- Main article: Binomial distribution
One is often interested in knowing how often one will observe in a sequence of revealed cards. This is given by simply counting: Given successive revealed cards, that is, given the set of all possible strings of length , the number of such strings that contain occurrences of is given by the binomial coefficient
If the probability of revealing Advanced Tech is given by , then the total probability of seeing a string of length with Advanced Tech cards is
- , where .
The probability measure thus defined is known as the binomial distribution.