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Geometry and probabilities

Grigoriou Giannis

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Summary

This thesis introduces three main areas: geometric probability, norm geometry,and Martingales' theory. These areas combine classical probability theory withmathematical and geometrical concepts.Geometric probability is the result of the need to understand random events,with Buffon's problem being a major milestone. This problem integratedgeometric parameters into probability theory. The classical definition ofprobability emerged through the work of Pierre de Fermat and Blaise Pascal,contributing to the development of the field.Norm geometry is another important area addressed in the paper. It combineseveryday experiences of distance with abstract mathematical thinking. Theconcepts of metric spaces and set distances are analyzed in detail, withexamples in Euclidean space to provide a better understanding of the field.The theory of Martingales is presented as a powerful tool of mathematicalanalysis, originating from gambling strategies. Martingales are used to examinesequences of random variables and find applications in various fields such aseconomics and computer science. Applications, such as stock price simulationand anomaly detection in time series, are presented using the Pythonprogramming language. The paper also highlights the perspectives ofstochastic analysis and artificial intelligence for future research.

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