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Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than ...
The law of large numbers is a fundamental concept in probability theory and statistics, tying together theoretical probabilities that we can calculate to the actual outcomes of experiments that we empirically perform. explained Business Insider
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. [1] These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. [2] There are several other (equivalent) approaches to formalising ...
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers .
A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states: If the events E n are independent and the sum of the probabilities of the E n diverges to infinity, then the probability that infinitely many of them occur is 1.
Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example (Kechris 1995, Sect. 17). This approach is based on the isomorphism theorem for standard Borel spaces ( Kechris 1995 , Theorem (15.6)).