Kolmogorov Probability Function Axiom

A Kolmogorov Probability Function Axiom is a probability function axiom that is part of the Kolmogorov's probability theory.

• Context:
  • It can define an Analytical Probability Function.
  • It was proposed by (Kolmogorov, 1933).
• Example(s):
  • Axiom₁ - [math]\displaystyle{ P(E)\in\mathbb{R}, P(E)\geq 0 \qquad \forall E\in F }[/math] (the probability value of a statistical event is a non-negative real number.)
  • Axiom₂ - [math]\displaystyle{ P(\Omega) = 1 }[/math]; The probability value of some elementary event is 1, and there is no elementary event outside of the sample space.
  • Axiom₃ - P(A ∪ B) = P(A) + P(B); a probability value is additive over distinct events.
  • Axiom₄ - if the Random Experiment is Continuous ...
  • …
• Counter-Example(s):
  • [math]\displaystyle{ P(\emptyset)=0 }[/math]; The probability of an Impossible Event (nothing happening) is 0.
  • Events A and B are independent iff P(A & B) = P(A)*P(B)
• See: Empirical Probability Function, Axiomatized System, Field of Statistics, Cox's Theorem, Probability Theory, Probability, Event (Probability Theory), Andrey Kolmogorov, Measure Space, Probability Space, Bayesian Theory.

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References

2018

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Probability_axioms Retrieved:2018-8-23.
  • In Kolmogorov's probability theory, the probability P of some event E, denoted [math]\displaystyle{ P(E) }[/math] , is usually defined such that P satisfies the Kolmogorov axioms, named after the Russian mathematician Andrey Kolmogorov, which are described below.

    These assumptions can be summarised as follows: Let (Ω, F, P) be a measure space with P(Ω) = 1. Then (Ω, F, P) is a probability space, with sample space Ω, event space F and probability measure P.

    An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.

1986

• (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
  • QUOTE: Consider a sample space, [math]\displaystyle{ S }[/math], and any event, [math]\displaystyle{ A }[/math], defined on S. If our experiment were performed one time, either [math]\displaystyle{ A }[/math] or A^C would be the outcome. If it were performed [math]\displaystyle{ n }[/math] times, the resulting set of sample outcomes would be members of [math]\displaystyle{ A }[/math] on [math]\displaystyle{ m }[/math] occasions, [math]\displaystyle{ m }[/math] being some integer between 0 and [math]\displaystyle{ n }[/math], inclusive. Hypothetically, we could continue this process an infinite number of times. As [math]\displaystyle{ n }[/math] gets large, the ratio m/n will fluctuate less and less (we will make that statement more precise a little later). The number that m/n convert to is called the empirical probability of [math]\displaystyle{ A }[/math] : that is, P(A) = lim_n→∞(m/n). … the very act of repeating an experiment under identical conditions an infinite number of times is physically impossible. And left unanswered is the question of how large [math]\displaystyle{ n }[/math] must be to give a good approximation for lim_n→∞(m/n).
  • The next attempt at defining probability was entirely a product of the twentieth century. Modern mathematicians have shown a keen interest in developing subjects axiomatically. It was to be expected, then, that probability would come under such scrutiny … The major breakthrough on this front came in 1933 when Andrei Kolmogorov published Grundbegriffe der Wahscheinlichkeitsrechnung (Foundations of the Theory of Probability.). Kolmogorov's work was a masterpiece of mathematical elegance - it reduced the behavior of the probability function to a set of just three or four simple postulates, three if the same space is limited to a finite number of outcomes and four if [math]\displaystyle{ S }[/math] is infinite.

1933

• (Kolmogorov, 1933) ⇒ Andrei Kolmogorov. (1933). “Grundbegriffe der Wahrscheinlichkeitsrechnung.” (Foundations of the Theory of Probability.). American Mathematical Society. ISBN:0828400237

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