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Concept# Probability theory

Summary

Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion).
Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major resul

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This course focuses on dynamic models of random phenomena, and in particular, the most popular classes of such models: Markov chains and Markov decision processes. We will also study applications in queuing theory, finance, project management, etc.

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This is an overview of a program of stochastic deformation of the mathematical tools of classical mechanics, in the Lagrangian and Hamiltonian approaches. It can also be regarded as a stochastic version of Geometric Mechanics.The main idea is to construct well defined probability measures strongly inspired by Feynman Path integral method in Quantum Mechanics. In contrast with other approaches, this deformation preserves the invariance under time reversal of the underlying classical (conservative) dynamical systems.

2012Let r : S x S -> R+ be the jump rates of an irreducible random walk on a finite set S, reversible with respect to some probability measure m. For alpha > 1, let g : N -> R+ be given by g(0) = 0, g(1) = 1, g(k) = (k/k - 1)(alpha), k >= 2. Consider a zero range process on S in which a particle jumps from a site x, occupied by k particles, to a site y at rate g(k)r(x, y). Let N stand for the total number of particles. In the stationary state, as N up arrow infinity, all particles but a finite number accumulate on one single site. We show in this article that in the time scale N1+alpha the site which concentrates almost all particles evolves as a random walk on S whose transition rates are proportional to the capacities of the underlying random walk.

2012Raphaël Gérard Théodore Michel Marie de Deloÿe et Fourcade de Fondeville

In 2019, Eliud Kipchoge ran a sub-two hour marathon wearing Nike's Alphafly shoes. Despite being the fastest marathon time ever recorded, it wasn't officially recognized as race conditions were tightly controlled to maximize his success. Besides, Kipchoge's use of Alphafly shoes was controversial, with some experts claiming that they might have provided an unfair competitive advantage. In this work, we assess the potential influence of advanced footwear technology and the likelihood of a sub-two hour marathon in official races, by studying the evolution of running top performances from 2001 to 2019 for long distances ranging from 10 km to marathon. The analysis is performed using extreme value theory, a field of statistics dealing with analysis of rare events. We find a significant evidence of performance-enhancement effect with a 10% increase of the probability that a new world record for marathon-men discipline is set in 2021. However, results suggest that achieving a sub-two hour marathon in an official race in 2021 is still very unlikely, and exceeds 10% probability only by 2025.