Editing Bernoulli process (section)

==Formal definition==
The Bernoulli process can be formalized in the language of [[probability space]]s as a random sequence  of independent realisations of a random variable that can take values of heads or tails. The state space for an individual value is denoted by <math>2=\{H,T\} .</math>

===Borel algebra===
Consider the [[countably infinite]] [[direct product]] of copies of <math>2=\{H,T\}</math>.  It is common to examine either the one-sided set <math>\Omega=2^\mathbb{N}=\{H,T\}^\mathbb{N}</math> or the two-sided set <math>\Omega=2^\mathbb{Z}</math>. There is a natural [[topology]] on this space, called the [[product topology]]. The sets in this topology are finite sequences of coin flips, that is, finite-length [[string (computer science)|strings]] of ''H'' and ''T'' (''H'' stands for heads and ''T'' stands for tails), with the rest of (infinitely long) sequence taken as "don't care".  These sets of finite sequences are referred to as [[cylinder set]]s in the product topology.  The set of all such strings forms a [[sigma algebra]], specifically, a [[Borel algebra]].  This algebra is then commonly written as <math>(\Omega, \mathcal{B})</math> where the elements of <math>\mathcal{B}</math> are the finite-length sequences of coin flips (the cylinder sets).

===Bernoulli measure===
If the chances of flipping heads or tails are given by the probabilities <math>\{p,1-p\}</math>, then one can define a natural [[measure (mathematics)|measure]] on the product space, given by <math>P=\{p, 1-p\}^\mathbb{N}</math> (or by <math>P=\{p, 1-p\}^\mathbb{Z}</math> for the two-sided process).  In another word, if a [[discrete random variable]] ''X'' has a ''Bernoulli distribution'' with parameter ''p'', where 0 ≤ ''p'' ≤ 1, and its [[probability mass function]] is given by  

:<math>pX(1)=P(X=1)=p</math> and <math>pX(0)=P(X=0)=1-p</math>.  

We denote this distribution by Ber(''p'').<ref name=":0" /> 

Given a cylinder set, that is, a specific sequence of coin flip results <math>[\omega_1, \omega_2,\cdots\omega_n]</math> at times <math>1,2,\cdots,n</math>, the probability of observing this particular sequence is given by 

:<math>P([\omega_1, \omega_2,\cdots ,\omega_n])= p^k (1-p)^{n-k}</math>
where ''k'' is the number of times that ''H'' appears in the sequence, and ''n''−''k'' is the number of times that ''T'' appears in the sequence. There are several different kinds of notations for the above; a common one is to write 
:<math>P(X_1=x_1, X_2=x_2,\cdots, X_n=x_n)= p^k (1-p)^{n-k}</math>
where each <math>X_i</math> is a binary-valued [[random variable]] with <math>x_i=[\omega_i=H]</math> in [[Iverson bracket]] notation, meaning either <math>1</math> if <math>\omega_i=H</math> or <math>0</math> if <math>\omega_i=T</math>.  This probability <math>P</math> is commonly called the [[Bernoulli measure]].<ref name=klenke>{{cite book |first=Achim |last=Klenke |title=Probability Theory |year=2006 |publisher=Springer-Verlag |isbn=978-1-84800-047-6}}</ref>

Note that the probability of any specific, infinitely long sequence of coin flips is exactly zero; this is because <math>\lim_{n\to\infty}p^n=0</math>, for any <math>0\le p<1</math>. A probability equal to 1 implies that any given infinite sequence has [[measure zero]].  Nevertheless, one can still say that some classes of infinite sequences of coin flips are far more likely than others, this is given by the [[asymptotic equipartition property]].

To conclude the formal definition, a Bernoulli process is then given by the probability triple <math>(\Omega, \mathcal{B}, P)</math>, as defined above.