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In probability theory, the notion of '''probability space'''
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is the conventional mathematical model of randomness.
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.
It formalizes three interrelated ideas by three mathematical notions.
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 sample point.
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 σ-algebra (called also σ-field) <math>\textstyle \mathcal F </math>
of subsets of <math>\textstyle \Omega</math>
(these subsets are called 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 set <math> \Omega = \{ \omega_1, \dots, \omega_n \} </math> of all sample points is called the '''sample space'''. Every subset <math> A \subset \Omega </math> of the sample space is called an '''event'''; its probability is the sum of probabilities of its sample points. For example, if <math> A = \{ \omega_1, \omega_8, \omega_9 \} </math> then <math> \mathbb{P} (A) = p_1 + p_8 + p_9 </math>.
 
A '''random variable''' <math> X </math> is described by real numbers <math> x_1, \dots, x_n </math> (not necessarily different) corresponding to the sample points <math> \omega_1, \dots, \omega_n. </math> Its '''expectation''' is <math> \mathbb{E} (X) = x_1 p_1 + \dots + x_n p_n. </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. For them, formulas (such as the addition rule, the multiplication rule, the inclusion-exclusion rule, the law of total probability, Bayes' rule etc.<ref>See [http://www.math.uah.edu/stat/prob/Probability.xhtml#Rules], [http://www.math.uah.edu/stat/prob/Conditional.xhtml#Properties]</ref>) are instrumental; probability spaces are not, they reign but do not rule.
 
''Explanation 1.''
Likewise, one may say that points are of no use in geometry.
Formulas connecting lengths and angles (such as Pythagorean theorem, law of sines etc.) are instrumental; points are not.
 
However, these useful formulas follow from the axioms of geometry formulated in terms of points (and some other notions).
It would be very cumbersome and unnatural, if at all possible,
to reformulate geometry avoiding points.
 
Similarly, the formulas of probability follow from the axioms of probability formulated in terms of probability spaces.
It would be very cumbersome and unnatural, if at all possible,
to reformulate probability theory avoiding probability spaces.
 
''Explanation 2.''
One of the most useful formulas is linearity of expectation:
<math> \mathbb{E} (aX+bY) = a \mathbb{E} (X) + b \mathbb{E} (Y) </math>
whenever <math> X, Y </math> are random variables and <math> a, b </math> are (non-random) coefficients. One may derive this formula avoiding probability spaces, by transforming the sum
: <math> \sum_{x,y} (ax+by) \mathbb{P} (X=x, Y=y ) </math>
into the linear combination
: <math> a \sum_x x \mathbb{P} (X=x) + b \sum_y y \mathbb{P} (Y=y). </math>
However, much better insight is provided by probability spaces: the expectation <math> \mathbb{E} (X) = x_1 p_1 + \dots + x_n p_n </math> is a linear function of the variables <math> x_1, \dots, x_n. </math> Moreover, a helpful connection to linear algebra appears: random variables form an <math>n</math>-dimensional linear space, and the expectation is a linear functional on this space.
 
==Notes==
<references />

Latest revision as of 03:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


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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