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Probability and Statistics

Probability and Statistics

The mathematics of uncertainty and randomness (probability) and of drawing sound conclusions from data (statistics). Probability theory traces to Pascal and Fermat's 1654 correspondence over a gambling dispute and was placed on a rigorous measure-theoretic footing by Andrey Kolmogorov in 1933; statistics developed largely independently, through Gauss's method of least squares and the hypothesis-testing framework built by Ronald Fisher, Jerzy Neyman and Egon Pearson in the early twentieth century.

Facts
Central Question
How can uncertainty, randomness and incomplete information themselves be reasoned about with mathematical precision, and what can data honestly be said to support? 2
Key Debate
The Bayesian versus frequentist divide over what a probability actually means, a rational agent's degree of belief, updated by evidence, against a long-run frequency of outcomes in repeated trials, a disagreement over a century old that still shapes how modern statistical inference is taught and practiced. 2
Classification
Pure or Applied
Both / Interdisciplinary 1
Connections

Associated With

Game Theory, Branches of Mathematics

Game theory shares its expected-value and optimization machinery with probability and statistics; mixed strategies are themselves probability distributions over pure strategies.

Source The Stanford Encyclopedia of Philosophy
Pierre-Simon Laplace, Mathematicians
Source Pierre-Simon Laplace, Biography (MacTutor History of Mathematics)

Includes

Bayes' Theorem, Theorems
Source MacTutor History of Mathematics Archive
Birkhoff Ergodic Theorem, Theorems
Borel-Cantelli Lemma, Theorems
Carl Friedrich Gauss, Mathematicians
Source MacTutor History of Mathematics Archive
Central Limit Theorem, Theorems
Source MacTutor History of Mathematics Archive
Chebyshev's Inequality, Theorems
Cochran's Theorem, Theorems
Cramer-Rao Inequality, Theorems
Cramer's Theorem (Large Deviations), Theorems
De Moivre-Laplace Theorem, Theorems
Donsker's Theorem, Theorems
Doob's Martingale Convergence Theorem, Theorems
Fisher-Neyman Factorization Theorem, Theorems
Gauss-Markov Theorem, Theorems
Glivenko-Cantelli Theorem, Theorems
Kolmogorov Extension Theorem, Theorems
Kolmogorov's Zero-One Law, Theorems
Markov's Inequality, Theorems
Neyman-Pearson Lemma, Theorems
Optional Stopping Theorem, Theorems
Paul Erdos, Mathematicians
Source MacTutor History of Mathematics Archive
Pierre de Fermat, Mathematicians
Source MacTutor History of Mathematics Archive
Probability, Concepts
Random Variable, Concepts
Rao-Blackwell Theorem, Theorems
Slutsky's Theorem, Theorems
Strong Law of Large Numbers, Theorems
Weak Law of Large Numbers, Theorems
Sources
1. Statistics (Wikipedia)
Lead section, applications
Quote, Lead section, applications
Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology.
View the Source
2. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statistics
  • Includes: Central Limit Theorem
  • Includes: Bayes' Theorem
View the Source
The Stanford Encyclopedia of Philosophy
Center for the Study of Language and Information, Stanford UniversityAssociated With: Game TheoryView the Source
Pierre-Simon Laplace, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsAssociated With: Pierre-Simon Laplace, https://mathshistory.st-andrews.ac.uk/Biographies/Laplace/
Quote, Associated With: Pierre-Simon Laplace, https://mathshistory.st-andrews.ac.uk/Biographies/Laplace/
put the theory of mathematical probability on a sound footing
View the Source
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