theory of probability and measure, a sub-probability measure is a measure that is closely related to probability measures. While probability measures always...

sub-probability) measures may not necessarily converge vaguely to a true probability measure, but rather to a sub-probability measure (a measure such...

In probability theory, a probability space or a probability triple ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} is a mathematical construct...

In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome...

{\displaystyle \Sigma _{*}} and probability measure P {\displaystyle \mathbb {P} } if Σ∗ is a filtration of the underlying probability space (Ω, Σ,  P {\displaystyle...

information theory, perplexity is a measure of uncertainty in the value of a sample from a discrete probability distribution. The larger the perplexity...

sense, this differs much from the modern meaning of probability, which in contrast is a measure of the weight of empirical evidence, and is arrived at...

}(A_{i}).} In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is σ-sub-additive....

properties that a risk measure might or might not have. A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity...

subsets of the finite measures are the sub-probability measures, which form a convex subset, and the probability measures, which are the intersection of the...

with these properties are called σ-algebras. Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences...

In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are totally ordered collections of subsets that are used to...

statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines...

B ↦ κ ( B , x 0 ) {\displaystyle B\mapsto \kappa (B,x_{0})} is a probability measure on ( Y , B ) {\displaystyle (Y,{\mathcal {B}})} In other words it...

distribution is a probability distribution that has a larger variance than a Poisson distribution with the same mean. Conversely, a sub-Poissonian distribution...

This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across...

substochastic kernel, sub-probability kernel or a sub-Markov kernel if all κ s {\displaystyle \kappa _{s}} are sub-probability measures a Markov kernel, stochastic...

a companion notion to the concept of measure zero, and is analogous to the notion of almost surely in probability theory. More specifically, a property...

Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically...

Space of Probability Measures, ETH Zürich, Birkhäuser Verlag, Basel, ISBN 3-7643-2428-7 Athreya, Krishna B.; Lahiri, Soumendra N. (2006), Measure theory...

its modern definition using sub-σ-algebras. If A is an event in F {\displaystyle {\mathcal {F}}} with nonzero probability, and X is a discrete random...

In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence...

Financial Engineers. Her career has spanned several sub-disciplines including catastrophic insurance, probability theory, and the finance of commodities. Her...

\right)^{\frac {1}{p}}} where ν is any probability distribution and h any ν-measurable function. Let μ be any measure, and ν the distribution whose density...

version of the Jaccard distance for measures, including probability measures. If μ {\displaystyle \mu } is a measure on a measurable space X {\displaystyle...

\operatorname {E} [\varphi (X)].} In this probability setting, the measure μ is intended as a probability P {\displaystyle \operatorname {P} } , the...

contrast, sub-Poissonian light requires the quantization of the electromagnetic field for a proper description and thus is a direct measure of the particle...

In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent...

a straightforward way to more general spaces, measure spaces, such as those that arise in probability theory. The term Lebesgue integration can mean...

probability proportional to size ('PPS') sampling, in which the selection probability for each element is set to be proportional to its size measure,...