- Bernoulli process
In

probability andstatistics , a**Bernoulli process**is a discrete-timestochastic process consisting ofa sequence of independentrandom variable s taking values over two symbols. Prosaically, a Bernoulli process iscoin flipping , possibly with an unfair coin. A variable in such a sequence may be called a**Bernoulli variable**.**Definition**A

**Bernoulli process**is a discrete-timestochastic process consisting of a finite or infinite sequence of independentrandom variable s "X"_{1}, "X"_{2}, "X"_{3},..., 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".In other words, a Bernoulli process is a sequence of independent identically distributed

Bernoulli trial s. 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 calledBernoulli trial s.Independence of

Bernoulli trial s 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

binomial distribution ;

*The number of trials needed to get "r" successes; this has anegative binomial distribution .

*The number of trials needed to get one success; this has ageometric 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 .**Formal definition**The Bernoulli process can be formalized in the language of

probability space s. A Bernoulli process is then aprobability space $(Omega,\; Pr)$ together with arandom variable "X" over the set $\{0,1\}$, so that for every $omega\; inOmega$, one has $X\_i(omega)=1$ with probability "p" and $X\_i(omega)=0$ with probability 1-"p".**Bernoulli sequence**Given a Bernoulli process defined on a

probability space $(Omega,\; Pr)$, then associated with every $omega\; in\; Omega$ is asequence of integers:$mathbb\{Z\}^omega\; =\; \{nin\; mathbb\{Z\}\; :\; X\_n(omega)\; =\; 1\; \}$

which is called the

**Bernoulli sequence**. So, for example, if $omega$ represents a sequence of coin flips, then the Bernoulli sequence is the list of integers for which the coin toss came out "heads".Almost all Bernoulli sequences areergodic sequence s.**Bernoulli map**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 theunit interval : in other words, the real numbers are distributed uniformly on the unit interval.The

shift operator "T" taking each random variable to the next,:$TX\_i=X\_\{i+1\}$

is then given by the

**Bernoulli map**or the2x mod 1 map :$b(z)=2z-lfloor\; 2z\; floor$

where $zin\; [0,1]$ represents a given sequence of measurements, and $lfloor\; z\; floor$ 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; theeigenvalue s are multiples of 1/2, and theeigenfunction s are theBernoulli polynomials .**Generalizations**The generalization of the Bernoulli process to more than two possible outcomes is called the

Bernoulli scheme .**References*** 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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