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Topic: Probability measures

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	probability. The Columbia Encyclopedia, Sixth Edition. 2001-05
		The probability measure of an event is sometimes defined as the ratio of the number of outcomes.
		The probability that either of the two events A and B will occur is given by the sum of their separate probabilities minus the probability that they will both occur.
		Thus if the probability that a certain man will live to be 70 is 0.5, and the probability that his wife will live to be 70 is 0.6, the probability that they will both live to be 70 is 0.5×0.6=0.3, and the probability that either the man or his wife will reach 70 is 0.5+0.6-0.3=0.8.
	www.bartleby.com /65/pr/probabil.html (667 words)

	Probability - Wikipedia, the free encyclopedia
		Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events.
		Probability applications include even more than statistics, which is usually based on the idea of probability distributions and the central limit theorem.
		Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable effect on the population as a whole, statistically.
	en.wikipedia.org /wiki/Probability (2712 words)

	Mean, Variance and Distributions
		Two measures of the prospects provided by such a portfolio are assumed to be sufficient for evaluating its desirability: the expected or mean value at the end of the accounting period and the standard deviation or its square, the variance, of that value.
		The probability that the actual outcome will be less than or equal to 20 is (0.20+0.30), or 0.50, and the probability that the outcome will be less than or equal to 30 is 1.00.
		Thus the probability of a shortfall is 0.3.
	www.stanford.edu /~wfsharpe/mia/rr/mia_rr1.htm (3358 words)

	Probability - Open Encyclopedia (Site not responding. Last check: 2007-10-17)
		In Cox's formulation, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions.
		The probability of an event is generally represented as a real number between 0 and 1.
		Note that a proper definition requires measure theory which provides means to cancel out those cases where the above limit does not provide the "right" result or is even undefined by showing that those cases have a measure of zero.
	www.open-encyclopedia.com /Probability (2628 words)

	Learn more about Probability in the online encyclopedia. (Site not responding. Last check: 2007-10-17)
		Another way probabilities are expressed is "odds", where the two numbers used represent the relative likelihood of the target event and the likelihood of all events other than the target event.
		A major impact of probability theory on everyday life is in risk assessment and in trade on commodity markets.
		Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable impact on the population as a whole, statistically.
	www.onlineencyclopedia.org /p/pr/probability.html (1648 words)

	Summary Measures of Uncertainty (Site not responding. Last check: 2007-10-17)
		Measuring uncertainty or information means assigning a number or a value from some ordinal scale to a given model of an epistemic state (in our case imprecise probability).
		The measurement of nonspecificity was settled relatively easily by a measure generalizing the Hartley measure [17].
		Lamata and Moral [49] were the first to propose the sum of the measure of nonspecificity and entropy-like measure as a measure of total uncertainty.
	ippserv.rug.ac.be /documentation/summary_measures/summary_measures.html (3740 words)

	Definition
		The two conditions from the definition of probability imply the axioms in the definition of measure.
		As a measure, probability is in fact a function with certain properties, defined on the field of events generated by an experiment.
		Probability as a number is in fact a limit, respectively the limit of the sequence of relative frequencies of occurrences of the event to measure, within a sequence of independent experiments.
	probability.infarom.ro /definition1.html (618 words)

	probability-measures-html
		Probability is introduced to judge two probability measures are
		This is stated as the Identitification Lemma for Probabilities:
		state some basic tools from measure theory that are very useful.
	mathcs.emory.edu /~jgardn3/test/originalnotes/probability-measures-html/index.html (81 words)

	PROBABILITY (Site not responding. Last check: 2007-10-17)
		A similar approach is applied to the existence of invariant probabilities for Markov processes on a metric space.
		We characterize the limit in the Birkhoff and Chacon-Ornstein ergodic theorems in terms of the limits of the expected occupations measures.
		This criterion is stated in terms of the type of convergence of the sequence of expected occupation measures.
	www.laas.fr /~lasserre/proba.html (405 words)

	Measuring the Workings of a cEA
		We will introduce here some statistical measures that will be of especial interest for analyzing the mode of operation of cellular evolutionary algorithms.
		The transition function of the evolutionary process, in turn based on the definition of the genetic operators, defines a sequence of probability measures over the generations.
		is the probability that two adjacent individuals (cells) have different genotypes, i.e., belong to two different blocks.
	neo.lcc.uma.es /cEA-web/measures.htm (513 words)

	Probability Measure
		Intuitively, the probability of an event is supposed to measure the long-term relative frequency of the event.
		, one of the fundamental theorems in probability.
		In particular, the inclusion-exclusion rule is as important in combinatorics (the study of counting measure) as it is in probability.
	www.math.uah.edu /stat/prob/Probability.xhtml (3011 words)

	Arunava Mukherjea --- Research Publications
		The convolution equation of Choquet and Deny and relatively invariant measures on semigroups (1971).
		Invariant measures and the converse of Haar's theorem on semi-topological semigroups (with Tserpes) (1972).
		Probability measures on semigroups of nonnegative matrices (with R. Darling) (1990).
	www.math.usf.edu /~arun/publications.html (1315 words)

	WebCab Probability and Statistics for Delphi v3.6
		The probability density function, cumulative distribution function and inverse, mean, variance, Skewness and Kurtosis are implemented where appropriate and/or their approximations for each distribution.
		Allows the fitting of linear and non-linear functions for a data set which may or may not exhibit measurement errors in accordance with the least squares approach.
		Kurtosis - Measures of whether the data set is peaked or flat realtive to the normal distribution.
	www.webcabcomponents.com /delphi/components/pss/index.shtml (1745 words)

	McGill SOCS: Seminar (Site not responding. Last check: 2007-10-17)
		While probabilities are quite popular, the AI literature is littered with different ways of representing uncertainty.
		There are sound and complete axiomatizations for the logic in the case that the underlying representation is (a) probability, (b) sets of probability measures, (c) belief functions, and (d) possibility measures.
		While the logic can be shown to be more expressive than the corresponding logic for reasoning about likelihood in the case of sets of probability measures (equi-expressive in the other cases), surprisingly, the satisfiability problem for the logic is NP-complete in all cases, no harder than satisfiability for propositional logic.
	www.cs.mcgill.ca /news/seminars/sem-2002_09_26.html (164 words)

	Chapter 5: Probabilism and Induction
		Given a coherent assignment of probabilities to a finite number of propositions, the probability of any further propositions is either determined or can be coherently assigned any value in a certain closed interval.
		Thus, the set of probability measures that assign the given values to the finite set of propositions must be convex.
		Probability logic need not tell you how to revise your opinion in such cases, any more than deductive logic need tell you which of an inconsistent set of premises to reject.
	www.princeton.edu /~bayesway/ProbThink/Chapter5.html (5363 words)

	Quantum Logic and Probability Theory (Stanford Encyclopedia of Philosophy)
		Mackey presents a sequence of six axioms, framing a very conservative generalized probability theory, that underwrite the construction of a ‘logic’ of experimental propositions, or, in his terminology, ‘questions’, having the structure of a sigma-orthomodular poset.
		Thus, a state is a consistent assignment of a probability weight to each test -- consistent in that, where two distinct tests share a common outcome, the state assigns that outcome the same probability whether it is secured as a result of one test or the other.
		The decision to accept measurements and their outcomes as primitive concepts in our description of physical systems does not mean that we must forgo talk of the physical properties of such a system.
	plato.stanford.edu /entries/qt-quantlog (8000 words)

	Comparison between possibility and probability
		The possibility functions are more natural for the representation of the feeling of incertitude: very precise information is not expected absolutely from someone, but we do hope for the greatest possible coherence in his remarks.
		Probability on the one hand, and possibility-necessity on the other hand, correspond to two extreme, therefore ideal situations.
		Moreover, the values of measures and distributions are selected with more freedom in the case of possibility than in the case of probability.
	www.survey.ntua.gr /main/labs/rsens/DeCETI/IRIT/MSI-FUSION/node98.html (203 words)

	Combinatorics and Probability - Numericana
		The first term is the probability the contestant was right and can't switch, whereas the second term is the probability he was wrong and could switch.
		If n independent random samples are either equal to 1 (with probability p) or to 0 (with probability 1-p), the sum of their values is a random variable which is said to have a binomial distribution; its average is np and its variance is np(1-p).
		Since all points on the surface of a sphere of radius R are equivalent, the average distance between two random points is the same as the average distance from a fixed point to a random one (assuming, of course, that the probability of a set is proportional to its spherical area).
	home.att.net /~numericana/answer/counting.htm (9900 words)

	Distance in Probability Measure Space
		Ideally, one would compare two measures on the basis of their assignment of probability to blocks of all sizes.
		Since the measures we will discuss are 1- and 2-step Markov measures, we opt to compare these measures on the basis of the 1- and 2-block probabilities which define them.
		One may anticipate that (in the limit of large class size) the invariant measure of such a process should be close to the fixed-point measure of the system of equations which define the class.
	www.santafe.edu /~hag/class/node27.html (515 words)

	Quantum haystacks
		Our probability is slightly different from classical probability, the same for logic; we end up with quantum logic and quantum probability.A commitment to this kind of mathematical structure, with which to model objects and processes in IR, depends on two critical assumptions.
		One may summarise this theorem by saying that the probability of a subspace is given by a simple algorithm derived from a projection onto the subspace and a special kind of operator, namely a statistical operator, or density matrix.
		In the case of evaluation, one starts with a relational conjoint structure and imposes some constraints given by what is to be measured, one then constructs a numerical representation of this structure leading to such measures as F (or E).
	portal.acm.org /citation.cfm?doid=1148170.1148171 (986 words)

	Definition of Algorithmic probability
		The algorithmic probability of any given finite output prefix q is the sum of the probabilities of the programs that compute something starting with q.
		Algorithmic probability is the main ingredient of Ray Solomonoff's theory of inductive inference, the theory of prediction based on observations.
		Algorithmic probability is closely related to the concept of Kolmogorov complexity.
	www.wordiq.com /definition/Algorithmic_probability (279 words)

	Universality and Complexity in Cellular Automata (1984)
		For cellular automata with translation invariant initial probability measures, stronger constraints may be obtained (analogous to those for ``stationary'' processes in communication theory [8]).
		For smooth probability measures on the ensemble of possible initial configurations, the results obtained in these two ways are almost always the same.
		in general depends on the probability measure for initial configurations, although for class 3 cellular automata, it is typically the same for at least a large class of smooth measures.
	www.stephenwolfram.com /publications/articles/ca/84-universality/5/text.html (2471 words)

	November 2002 Actuarial Review - Setting Capital Requirements With Coherent Measures of Risk- Part 2
		The supremum of the expected losses over this class of probability measures is the average of the n largest losses.
		To calculate a measure of risk with the Wang Transform, you first arrange the possible values of X in increasing order, calculate the cumulative probabilities, and then calculate the transform using the above formula.
		The measure of risk is a maximum of the expected values over two probability measures.
	www.casact.org /pubs/actrev/nov02/latest.htm (780 words)

	Unconditional Probability
		An unconditional probability is the independent chance that a single outcome results from a sample of possible outcomes.
		To find the unconditional probability of an event, sum the outcomes of the event and divide by the total number of possible outcomes.
		Conditional probability measures the chance of an occurrence ignoring any knowledge gained from previous or external events.
	www.investopedia.com /terms/u/unconditional_probability.asp (403 words)

	mp_arc 01-208 (Site not responding. Last check: 2007-10-17)
		Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set.
		Let T be a Lasota-Yorke map on the interval I, let Y be a non trivial sub-interval of I and g, be a strictly positive potential which belongs to BV and admits a conformal measure m.
		m) conditionally invariant probability measures to non absorption in Y. These conditions imply also existence of an invariant probability measure on the set X of points which never fall into Y. Our conditions allow rather ``large'' holes.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=01-208 (102 words)

	OUP: UK General Catalogue
		With this fine exposition, the author gives a cohesive account of the theory of probability measures on complete metric spaces (which he views as an alternative approach to the general theory of stochastic processes).
		After a general description of the basics of topology on the set of measures, he discusses regularity, tightness, and perfectness of measures, properties of sampling distributions, and metrizability and compactness theorems.
		Next, he describes arithmetic properties of probability measures on metric groups and locally compact abelian groups.
	www.oup.com /uk/catalogue/?ci=9780821838891 (340 words)

	Homeland Security Presidential Directive-3
		Protective Measures are the specific steps an organization shall take to reduce its vulnerability or increase its ability to respond during a period of heightened alert.
		It is recognized that departments and agencies may have several preplanned sets of responses to a particular Threat Condition to facilitate a rapid, appropriate, and tailored response.
		Institutionalizing a process to assure that all facilities and regulated sectors are regularly assessed for vulnerabilities to terrorist attacks, and all reasonable measures are taken to mitigate these vulnerabilities.
	www.whitehouse.gov /news/releases/2002/03/20020312-5.html (1531 words)

	Abstracts of Joseph Y. Halpern's Publications
		The measurable case is essentially a formalization of (the propositional fragment of) Nilsson's probabilistic logic.
		For example, the measure of belief in an event turns out to be represented by an interval (defined by the inner and outer measure), rather than by a single number.
		Probability measures, ranking functions, possibility measures, and (under the appropriate definitions) sets of probability measures can all be viewed as defining algebraic conditional plausibility measures.
	www.cs.cornell.edu /home/halpern/abstract.html (17597 words)

	Probability Spaces
		In this chapter, we study the paradigm of the random experiment and its mathematical model, the probability space.
		The main objects in this model are sample spaces, events, random variables, and probability measures.
		We also study several concepts of fundamental importance: conditional probability and independence, and we study several modes of convergence that are appropriate for random variables.
	www.math.uah.edu /stat/prob (113 words)

	Edwards: On the existence of probability measures with given marginals
		Edwards: On the existence of probability measures with given marginals
		On the existence of probability measures with given marginals.
		STRASSEN, The existence of probability measures with given marginals, Ann.
	www.numdam.org /item?id=AIF_1978__28_4_53_0 (187 words)

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