Bernoulli process

(win, and there is Bernoulli British: Bernoulli process) is a stochastic process of the disintegration time consisting of independent random variable lines taking two values in Bernoulli process. The Bernoulli process is, so to speak, a coin toss, and the coin may not be necessarily a fair thing. I call the random variable in such a stochastic process Bernoulli variable (Bernoulli variable).

Table of contents

Definition

The Bernoulli process is limitedness or infinite independent random variable line X₁, X₂, X₃, for a stochastic process of the disintegration time... からなる. About this random variable line, next is managed.

• About each i, a value of X_i is 0 or 1.
• About all values of i, probability p becoming Xi = 1 is the always same.

In other words, it is the line of the Bernoulli trial that the Bernoulli process becomes independent, and probability distribution is the same. I may call two values that I can take of individual X_i "success, success" and "failure, failure". When I was expressed with 0 or 1, as for the value, it may be said even if I express the success number of times about "the trial" of the i joint. Variable X_i of individual success / failure is called Bernoulli trial, too.

The attribute of the memoryless nature is included in the independency of the Bernoulli trial, too. In other words, the result of the past trial brings no information about a future result. As for the future trial from any point in time, Bernoulli trial is independent for the past (I call this a fresh start attribute).

There is the following characteristic in the random variable in the Bernoulli process.

• Of the beginning the success number of times in the の trial is binominal distribution n times.
• The trial number of times necessary to get the success of the r time is negative binominal distribution.
• The trial number of times necessary to get one success is geometric distribution, and this is a special case of the negative binominal distribution.

I call a problem to identify the property in the Bernoulli process for the cause only with the specimen of the Bernoulli trial of the limited unit with "checking if a coin is fair" (the issue of equitableness of the coin).

Formal definition

The Bernoulli process is formalized by a language of the probability space. In Bernoulli process, I gather ${\displaystyle 0,1}$  Probability space with random variable X where に relates to ${\displaystyle (\Omega, Pr)}$  Of であり, all ${\displaystyle \omega \in \Omega}$  It is and is probability p ${\displaystyle X_{i}(\omega) = 1}$  In neighbor, probability 1-p ${\displaystyle X_{i}(\omega) = 0}$  となる.

Bernoulli line

Probability space ${\displaystyle (\Omega, Pr)}$  When there is Bernoulli process defined by the top,${\displaystyle \omega \in \Omega}$  The line of the integer of every に next supports.

${\displaystyle \mathbb {Z} ^{\omega} = n\in \mathbb {Z} :X_{n}(\omega) = 1}$ 

I call this a Bernoulli line (Bernoulli sequence). Therefore, for example,${\displaystyle \omega}$  But, the Bernoulli process expressed the result of the coin toss in the line of the integer when I expressed the line of the coin toss.

An almost all Bernoulli line is a エルゴード line.

Bernoulli map

Because all trials take one of two values, I can consider the line of the trial to have expressed a real number by binary number system. If probability p is 1/2, all binary lines are produced with the same probability, and the gauging of the complete addition group of the Bernoulli process is equivalent with the equality gauging in the unit section. In other words, those real numbers are distributed equally on the unit section.

Shift operator T gives next (random variable) of each random variable as follows.

${\displaystyle TX_{i}=X_{i+1}}$ 

This is given by the next Bernoulli map (Bernoulli map).

${\displaystyle b(z)=2z-\lfloor 2z\rfloor}$ 

Here ${\displaystyle z\in [0,1]}$  I express は 測定列,${\displaystyle \lfloor z\rfloor}$  I express は 床関数 (maximum integer, i.e., not to be beyond z). The Bernoulli map is equivalent to a decimal when I considered z to be binary expression essentially.

The Bernoulli map is an exact possible solution model of the conclusiveness chaos. transfer operator of the Bernoulli map is a possible solution. The eigenvalue is a multiple of 1/2, and the characteristic function is Bernoulli multinomial expression.

of Bernoulli line

I call the thing which generalized three Bernoulli processes to take values more than it with Bernoulli system.

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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