Non-additive probability
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Conditional and Non-Standard Probability
In the classical theory of probability we start by a pair (Ω, ), with Ω set of elementary events and σ-algebra of subsets of Ω, called events. A probability on is defined as a countable additive and positive function p such that p(Ω)=1. If A and B are two events, the conditional probability p(A/B) is defined if and only if p(B) is not null, by considering the quotient p(AB)/p(B). In the theory ...
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Non-additive probability goes back to the very beginning of probability theory— the work of Jacob Bernoulli. Bernoulli’s calculus for combining arguments allowed both sides of a question to attain only small or zero probability, and he also thought the probabilities for two sides might sometimes add to more than one (Shafer 1978). Twentieth-century non-additive probability has roots in both mat...
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Based on a new analytical approach to the definition of additive free con-volution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory.
متن کاملNonconglomerability for countably additive Measures that are not κ-additive
Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-‐ additive has conditional probabilities that fail to be conglomerable i...
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تاریخ انتشار 2002