### Probabilistic Symmetries and Invariance Principles (Probability and its Applications)

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The earliest known forms of probability and statistics were developed by Arab mathematicians studying cryptography between the 8th and 13th centuries. Al-Khalil — wrote the Book of Cryptographic Messages which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels. Al-Kindi — made the earliest known use of statistical inference in his work on cryptanalysis and frequency analysis.

An important contribution of Ibn Adlan — was on sample size for use of frequency analysis.

The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes [15]. Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal Christiaan Huygens gave the earliest known scientific treatment of the subject.

The theory of errors may be traced back to Roger Cotes 's Opera Miscellanea posthumous, , but a memoir prepared by Thomas Simpson in printed first applied the theory to the discussion of errors of observation. Simpson also discusses continuous errors and describes a probability curve. The first two laws of error that were proposed both originated with Pierre-Simon Laplace.

The first law was published in and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error, disregarding sign. The second law of error was proposed in by Laplace and stated that the frequency of the error is an exponential function of the square of the error.

Daniel Bernoulli introduced the principle of the maximum product of the probabilities of a system of concurrent errors. He gave two proofs, the second being essentially the same as John Herschel 's Donkin , , and Morgan Crofton Peters 's formula [ clarification needed ] for r , the probable error of a single observation, is well known.

Augustus De Morgan and George Boole improved the exposition of the theory. Andrey Markov introduced [22] the notion of Markov chains , which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov Like other theories , the theory of probability is a representation of its concepts in formal terms—that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.

There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation see probability space , sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem , 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. In both cases, the laws of probability are the same, except for technical details.

There are other methods for quantifying uncertainty, such as the Dempster—Shafer theory or possibility theory , but those are essentially different and not compatible with the laws of probability as usually understood. Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions.

Governments apply probabilistic methods in environmental regulation , entitlement analysis Reliability theory of aging and longevity , and financial regulation. A good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole.

An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict. In addition to financial assessment, probability can be used to analyze trends in biology e. As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring and can assist with implementing protocols to avoid encountering such circumstances.

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Probability is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play. The discovery of rigorous methods to assess and combine probability assessments has changed society. Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure.

Failure probability may influence a manufacturer's decisions on a product's warranty. The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

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Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a dice can produce six possible results. One collection of possible results gives an odd number on the dice. These collections are called "events". If the results that actually occur fall in a given event, the event is said to have occurred.

To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events events with no common results, e.

## Probabilistic Symmetries and Invariance Principles | SpringerLink

See Complementary event for a more complete treatment. If two events, A and B are independent then the joint probability is. If either event A or event B but never both occurs on a single performance of an experiment, then they are called mutually exclusive events. Conditional probability is the probability of some event A , given the occurrence of some other event B.

It is defined by [32].

### Probabilistic Symmetries and Invariance Principles

In this form it goes back to Laplace and to Cournot ; see Fienberg See Inverse probability and Bayes' rule. In a deterministic universe, based on Newtonian concepts, there would be no probability if all conditions were known Laplace's demon , but there are situations in which sensitivity to initial conditions exceeds our ability to measure them, i. In the case of a roulette wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty though as a practical matter, this would likely be true only of a roulette wheel that had not been exactly levelled — as Thomas A.

Bass' Newtonian Casino revealed.

This also assumes knowledge of inertia and friction of the wheel, weight, smoothness and roundness of the ball, variations in hand speed during the turning and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of a roulette wheel. Physicists face the same situation in kinetic theory of gases, where the system, while deterministic in principle , is so complex with the number of molecules typically the order of magnitude of the Avogadro constant 6.

Probability theory is required to describe quantum phenomena. The objective wave function evolves deterministically but, according to the Copenhagen interpretation , it deals with probabilities of observing, the outcome being explained by a wave function collapse when an observation is made. However, the loss of determinism for the sake of instrumentalism did not meet with universal approval. Albert Einstein famously remarked in a letter to Max Born : "I am convinced that God does not play dice". From Wikipedia, the free encyclopedia.

## Probabilistic Symmetries And Invariance Properties

For the mathematical field of probability specifically rather than a general discussion, see Probability theory. For other uses, see Probability disambiguation. Not to be confused with Probably. Main article: Probability interpretations.

Further information: Likelihood. Main article: History of probability. Further information: History of statistics. What's New - Home - Login.

### Probabilistic Symmetries and Invariance Principles

School Donation Program In Memory of How To Swap Books? Olav Kallenberg , is a probability theorist known for his work on exchangeable stochastic processes and for his graduate-level textbooks and monographs. During , Kallenberg served as the Editor-in-Chief of Probability Theory and Related Fields , one of the world's leading journals in probability.

He has worked as a probabilist in Sweden and in the United States. Kallenberg entered doctoral studies in mathematical statistics at KTH, but left his studies to work in operations analysis for a consulting firm in Gothenburg. After earning his doctoral degree, Kallenberg stayed with Chalmers as a lecturer. Kallenberg was appointed a full professor in Uppsala University.

United States Later he moved to the United States. Kallenberg is a Fellow of the Institute of Mathematical Statistics. Springer -Verlag, New York ISBN Kallenberg, O. Springer Series in Statistics. MR Scientific papers Homogeneity and the strong Markov property. Spreading and predictable sampling in exchangeable sequences and processes. Multiple integration with respect to Poisson and Levy processes with J. Fields , —