- Bernoulli process
probabilityand statistics, a Bernoulli processis a discrete-time stochastic processconsisting ofa sequence of independent random variables taking values over two symbols. Prosaically, a Bernoulli process is coin flipping, possibly with an unfair coin. A variable in such a sequence may be called a Bernoulli variable.
* 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 "i"th "trial".The individual success/failure variables "X""i" are also called Bernoulli trials.
Bernoulli trials implies memorylessness property: past trials do not provide any information regarding future outcomes. From any given time, future trials is also a Bernoulli process independent of the past (fresh-start property).
Random variables associated with the Bernoulli process include
*The number of successes in the first "n" trials; this has a
*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.
The problem of determining the process, given only a limited sample of Bernoulli trials, is known as the problem of
checking if a coin is fair.
The Bernoulli process can be formalized in the language of
probability spaces. A Bernoulli process is then a probability spacetogether with a random variable"X" over the set , so that for every , one has with probability "p" and with probability 1-"p".
Given a Bernoulli process defined on a
probability space, then associated with every is a sequenceof integers
which is called the Bernoulli sequence. So, for example, if represents a sequence of coin flips, then the Bernoulli sequence is the list of integers for which the coin toss came out "heads".
Almost allBernoulli sequences are ergodic sequences.
Because every trial has one of two possible outcomes, a sequence of trials may be represented by the binary digits of a
real number. When the probability "p" = 1/2, all possible distributions are equally likely, and thus the measure of the σ-algebra of the Bernoulli process is equivalent to the uniform measure on the unit interval: in other words, the real numbers are distributed uniformly on the unit interval.
shift operator"T" taking each random variable to the next,
is then given by the Bernoulli map or the
2x mod 1 map
where represents a given sequence of measurements, and is the
floor function, the largest integer less than "z". The Bernoulli map essentially lops off one digit of the binary expansion of "z".
The Bernoulli map is an exactly solvable model of deterministic chaos. The
transfer operator, or Frobenius-Perron operator, of the Bernoulli map is solvable; the eigenvalues are multiples of 1/2, and the eigenfunctions are the Bernoulli polynomials.
The generalization of the Bernoulli process to more than two possible outcomes is called the
* Carl W. Helstrom, "Probability and Stochastic Processes for Engineers", (1984) Macmillan Publishing Company, New York ISBN 0-02-353560-1.
* Dimitri P. Bertsekas and John N. Tsitsiklis, "Introduction to Probability", (2002) Athena Scientific, Massachusetts ISBN 1-886529-40-X
* Pierre Gaspard, "r-adic one-dimensional maps and the Euler summation formula", "Journal of Physics A", 25 (letter) L483-L485 (1992). "(Describes the eigenfunctions of the transfer operator for the Bernoulli map)"
* Dean J. Driebe, "Fully Chaotic Maps and Broken Time Symmetry", (1999) Kluwer Academic Publishers, Dordrecht Netherlands ISBN 0-7923-5564-4 "(Chapters 2, 3 and 4 review the Ruelle resonances and subdynamics formalism for solving the Bernoulli map)".
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