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Axiomatic probability theory deliberately avoids a definition of a random sequence.
[2] Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The
Bourbaki school considered the statement "let us consider a random sequence" an
abuse of language.
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Derive from sampling in [0,1] and can be projected back on any set by inverting the probability.
Random sampling in [0,1] is sampling a subsets partition^1 of [0,1] with a probability proportional to their length. Then inverting the probability (for ordered set) obtain the probability of a given set
^1 subdivision of [0,1] in a subinterevals, S, such that, for any a,B \in S A \cap B = \nullset
The concept of random sampling is important either in statistics and in probability. This mean that we can choose among a set its components “randomly”.
Randomly non necessarily literally means that there is no law (expressed in equations) or algorithm (expressed in actions or some programing code) that connects one pick to another. The pick of set elements do not have to be obtained by subsequent action: they can be concurrent instead. Besides, they do not need to be independent, in the sense of probabilistic dependence. They can depend on each other while this dependence does not imply causation (in the sense that one implies the other).
Random sampling is significant when the set of the domain is subdivided into disjoint parts: a partition. Therefore
Definition: Given a set (having the structure of a \sigma algebra) \Omega let a partition of \Omega, denoted as {\cal P} (\Omega):
{\cal P} (\Omega) = \{ x | \cup_{\cal P} x = \Omega \& \forall y,z \in \Omega y\cap z = \emptyset \}
Through probability P defined over {\cal P} (\Omega) each element x of the set is mapped into the closed interval [0,1] and it is guaranteed that P[ \cup_{\cal P} x] = 1.
There is not necessarily ordering in {\cal P} (\Omega) but we can arbitrarily arrange the set by associating each element of {\cal P} with a subset of [1,1] (we thus assume that the set has at most the same cardinality of the continuum \aleph_0.)
By arranging or re-arrainging the numbers in [0,1], we thus imply (since P is bijective) a re-arrangement of the set {\cal P} (\Omega).
Definition: we call sequence of elements in {\cal P} (\Omega), denoted as \cal S a denumerable set of elements in {\cal P} (\Omega):
{\cal S} = \{ x_1 \cdot \cdot \cdot \}
Definition: we call a sequence a random sequence if it ha no description shorter that itself via a universal Turing machine.
Theorem: A random sequence of integer through inverting P define a random sequence on the set {\cal P} (\Omega)
Prof. - The prof is trivial
Kolmogorov's definition of a random string was that it is random if has no description shorter than itself via a
universal Turing machine.
[9]However, investigating the ramndoness of strings composed by 0 and 1 to which at the end any number can betweeen 0 and 1can be conducted it appears that there can be several types of ramdomness because to generate one number, randomly, between 0 and 1, means generating a possibly infinite sequence of bit and repeating possibly infinite times the process.
Maybe it would be more fruitful to define a a-causal sequence, meaning for it a sequence for which not a causal explanation can be given.
First:
- Inevitable Randomness in Discrete Mathematics by József Beck 2009 ISBN 0-8218-4756-2 page 44
- Jump up^ Algorithms: main ideas and applications by Vladimir Andreevich Uspenskiĭ, Alekseĭ, Lʹvovich Semenov 1993 Springer ISBN 0-7923-2210-X page 166
- Church, Alonzo (1940). "On the Concept of Random Sequence". Bull. Amer. Math. Soc. 46: 130–136. doi:10.1090/S0002-9904-1940-07154-X.
Conditional probability is a selection rule that applies to the events set.
Pearl: