User:Boris Tsirelson/Sandbox1: Difference between revisions

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In probability theory, the conventional mathematical model of randomness is a '''probability space'''.
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It formalizes three interrelated ideas by three mathematical notions.
The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.
First, a sample point (called also elementary event), —
something to be chosen at random (outcome of experiment, state of nature, possibility etc.)
Second, an event, —
something that will occur or not, depending on the chosen elementary event.
Third, the [[probability]] of an event.


Alternative models of randomness (finitely additive probability, non-additive probability)
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
are sometimes advocated in connection to various probability interpretations.
 
==Introduction==
The notion "probability space" provides a basis of the formal structure of probability theory.
It may puzzle a non-mathematician for several reasons:
 
* it is called "space" but is far from geometry;
 
* it is said to provide a basis, but many people applying probability theory in practice neither understand nor need this quite technical notion.
 
These puzzling facts are explained below. First, a mathematical definition is given; it is quite technical, but the reader may skip it.
Second, an elementary case (finite probability space) is presented. Third, the puzzling facts are explained.
Next topics are countably infinite probability spaces, and general probability spaces.
 
==Definition==
A probability space is a [[measure (mathematics)|measure space]] such that the measure of the whole space is equal to 1.
 
In other words: a probability space is a triple <math>\textstyle (\Omega, \mathcal F, P)</math>
consisting of a [[set]] <math>\textstyle \Omega</math> (called the [[sample space]]),
a [[sigma-algebra|σ-algebra]] (called also σ-field) <math>\textstyle \mathcal F </math>
of subsets of <math>\textstyle \Omega</math>
(these subsets are called [[Event (probability theory)|events]]),
and a  [[measure (mathematics)|measure]] <math>\textstyle P</math> on <math>\textstyle (\Omega, \mathcal F)</math>
such that <math>\textstyle P(\Omega)=1</math> (called the probability measure).
 
== Elementary level: finite probability space ==
 
On the elementary level, a probability space consists of a finite number <math>n</math>
of sample points <math> \omega_1, \dots, \omega_n </math> and their probabilities
<math> p_1, \dots, p_n </math> --- positive numbers satisfying <math> p_1 + \dots + p_n = 1. </math>
 
== The puzzling facts explained ==
=== Why "space"? ===
''Fact:'' it is called "space" but is far from geometry.
 
''Explanation:'' see [[Space (mathematics)]].
 
=== What is it good for? ===
''Fact:'' it is said to provide a basis, but many people applying probability theory in practice
do not need this notion.
 
''Explanation 1.''
Likewise, one may say that points are of no use in geometry.
Numerous formulas, connecting lengths and angles, are instrumental;
points are not, they reign but do not rule.
 
However, axioms of geometry are formulated in terms of points (and some other notions).
It would be very cumbersome and unnatural, if at all possible,
to reformulate geometry without points.
 
Similarly, it would be very cumbersome and unnatural, if at all possible,
to reformulate probability theory without a probability space.

Latest revision as of 02:25, 22 November 2023


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The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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