Grundbegriffe der Wahrscheinlichkeitsrechnung¶
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer.
Cited by¶
8 citations across 8 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Conditional Probability
- Conditional probability is the probability of one event \(A\) relative to the assumption that another event \(B\) is known to have occurred — formally \(P(A \mid B) = P(A \cap B) / P(B)\), defined when \(P(B) > 0\).
This sourceThe measure-theoretic axiomatization of probability, defining conditional probability as P(A∩B)/P(B) and conditional expectation with respect to a sigma-algebra.
- Conditional probability is the probability of one event \(A\) relative to the assumption that another event \(B\) is known to have occurred — formally \(P(A \mid B) = P(A \cap B) / P(B)\), defined when \(P(B) > 0\).
- Law of Large Numbers
- The strong statement implies the weak; the weak alone permits a path that keeps returning, at ever rarer indices, to large excursions.
This sourceChapter VI treats both the ordinary and the strong law of large numbers, fixing the distinction between convergence in probability and almost-sure convergence of the empirical mean.
- The strong statement implies the weak; the weak alone permits a path that keeps returning, at ever rarer indices, to large excursions.
- Measure
- In probability a probability measure assigns total mass one to the sample space and additive masses to disjoint events — the Kolmogorov axioms are literally the measure axioms with a normalization constraint, so probability theory is, structurally, measure theory with total mass one.
This sourceAxiomatizes probability as a measure normalized to total mass one — the Kolmogorov axioms are the measure axioms (non-negativity, countable additivity) plus normalization.
- In probability a probability measure assigns total mass one to the sample space and additive masses to disjoint events — the Kolmogorov axioms are literally the measure axioms with a normalization constraint, so probability theory is, structurally, measure theory with total mass one.
- Nonparametric Methods
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Probability
- Probability is the calibrated quantification of uncertainty: a numerical assignment to events or propositions that obeys a stated set of coherence rules and supports consistent reasoning and decision-making under incomplete information, the formal commitment articulated by Kolmogorov (1933) in the measure-theoretic axiomatization of probability.
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Probability is the calibrated quantification of uncertainty: a numerical assignment to events or propositions that obeys a stated set of coherence rules and supports consistent reasoning and decision-making under incomplete information, the formal commitment articulated by Kolmogorov (1933) in the measure-theoretic axiomatization of probability.
- Set and Membership
- In statistics and probability, sample spaces and event spaces are sets, and probability measures are functions on the power set (or on a σ-algebra of measurable sets) following the Kolmogorov axiomatization
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- In statistics and probability, sample spaces and event spaces are sets, and probability measures are functions on the power set (or on a σ-algebra of measurable sets) following the Kolmogorov axiomatization
- Stationarity
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Stochasticity vs. Determinism
- Kolmogorov (1933) gave this distinction its modern measure-theoretic foundation.
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Kolmogorov (1933) gave this distinction its modern measure-theoretic foundation.
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