Kolmogorov Probability Function Axiom

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



References

2018

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 AC 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) = limn→∞(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 limn→∞(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