In probability and statistics, a Bernoulli process is a discrete-time stochastic process consisting of a 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. Probability is the likelihood that something is the case or will happen. ... This article is about the field of statistics. ... Discrete time is non-continuous time. ... In the mathematics of probability, a stochastic process is a random function. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... Coin flipping or coin tossing is the practice of throwing a coin in the air to resolve a dispute between two parties or otherwise choose between two alternatives. ...

Definition

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 Discrete time is non-continuous time. ... In the mathematics of probability, a stochastic process is a random function. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

• 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 ith "trial". The individual success/failure variables X_i are also called Bernoulli trials. In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ... In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ...

Independence of 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). In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ...

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.

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. In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... This article or section does not cite its references or sources. ...

Formal definition

The Bernoulli process can be formalized in the language of probability spaces. A Bernoulli process is then a probability space (Ω,Pr) together with a random variable X over the set {0,1}, so that for every , one has X_i(ω) = 1 with probability p and X_i(ω) = 0 with probability 1-p. In mathematics, the definition of the probability space is the foundation of probability theory. ... In mathematics, the definition of the probability space is the foundation of probability theory. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

Bernoulli sequence

Given a Bernoulli process defined on a probability space (Ω,Pr), then associated with every is a sequence of integers In mathematics, the definition of the probability space is the foundation of probability theory. ... For other senses of this word, see sequence (disambiguation). ...

which is called the Bernoulli sequence. So, for example, if ω 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 are ergodic sequences. In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties. ...

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 the unit interval: in other words, the real numbers are distributed uniformly on the unit interval. The binary or base-two numeral system is a system for representing numbers in which a radix of two is used; that is, each digit in a binary numeral may have either of two different values. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

The shift operator T taking each random variable to the next, In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...

TX_i = X_{i + 1}

is then given by the Bernoulli map or the 2x mod 1 map The dyadic transformation (also known as the dyadic map, 2x mod 1 map or Bernoulli map) is the mapping produced by the rule and for all n ≥ 0. ...

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 floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ...

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. For other uses, see Chaos Theory (disambiguation). ... In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. ... In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. ...

Generalizations

The generalization of the Bernoulli process to more than two possible outcomes is called the Bernoulli scheme. In mathematics, the Bernoulli scheme is a generalization of the Bernoulli process to more than two possible outcomes. ...

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).