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In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable additivity. [1] The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume ) is that a probability measure must ...
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The measure corresponding to a CDF is said to be induced by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies. The probability of a set in the σ-algebra is defined as
Bierlein's measure extension theorem is a result from measure theory and probability theory on extensions of probability measures. The theorem makes a statement about when one can extend a probability measure to a larger σ-algebra. It is of particular interest for infinite dimensional spaces.
The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics , mathematics , science and philosophy to draw conclusions about the likelihood of potential ...
When the underlying measure on (, ()) is finite, the distribution function in Definition 3 differs slightly from the standard definition of the distribution function (in the sense of probability theory) as given by Definition 2 in that for the former, = while for the latter, () = = ().
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also ...
A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique.