V.6 The Central Limit Theorem
by Imre Leader, June Barrow-Green, Timothy Gowers
The Princeton Companion to Mathematics
- Cover
- Title
- Copyright
- Contents
- Preface
- Contributors
- Part I Introduction
- Part II The Origins of Modern Mathematics
- Part III Mathematical Concepts
- III.1 The Axiom of Choice
- III.2 The Axiom of Determinacy
- III.3 Bayesian Analysis
- III.4 Braid Groups
- III.5 Buildings
- III.6 Calabi–Yau Manifolds
- III.7 Cardinals
- III.8 Categories
- III.9 Compactness and Compactification
- III.10 Computational Complexity Classes
- III.11 Countable and Uncountable Sets
- III.12 C*-Algebras
- III.13 Curvature
- III.14 Designs
- III.15 Determinants
- III.16 Differential Forms and Integration
- III.17 Dimension
- III.18 Distributions
- III.19 Duality
- III.20 Dynamical Systems and Chaos
- III.21 Elliptic Curves
- III.22 The Euclidean Algorithm and Continued Fractions
- III.23 The Euler and Navier-Stokes Equations
- III.24 Expanders
- III.25 The Exponential and Logarithmic Functions
- III.26 The Fast Fourier Transform
- III.27 The Fourier Transform
- III.28 Fuchsian Groups
- III.29 Function Spaces
- III.30 Galois Groups
- III.31 The Gamma Function
- III.32 Generating Functions
- III.33 Genus
- III.34 Graphs
- III.35 Hamiltonians
- III.36 The Heat Equation
- III.37 Hilbert Spaces
- III.38 Homology and Cohomology
- III.39 Homotopy Groups
- III.40 The Ideal Class Group
- III.41 Irrational and Transcendental Numbers
- III.42 The Ising Model
- III.43 Jordan Normal Form
- III.44 Knot Polynomials
- III.45 K-Theory
- III.46 The Leech Lattice
- III.47 L-Functions
- III.48 Lie Theory
- III.49 Linear and Nonlinear Waves and Solitons
- III.50 Linear Operators and Their Properties
- III.51 Local and Global in Number Theory
- III.52 The Mandelbrot Set
- III.53 Manifolds
- III.54 Matroids
- III.55 Measures
- III.56 Metric Spaces
- III.57 Models of Set Theory
- III.58 Modular Arithmetic
- III.59 Modular Forms
- III.60 Moduli Spaces
- III.61 The Monster Group
- III.62 Normed Spaces and Banach Spaces
- III.63 Number Fields
- III.64 Optimization and Lagrange Multipliers
- III.65 Orbifolds
- III.66 Ordinals
- III.67 The Peano Axioms
- III.68 Permutation Groups
- III.69 Phase Transitions
- III.70 π
- III.71 Probability Distributions
- III.72 Projective Space
- III.73 Quadratic Forms
- III.74 Quantum Computation
- III.75 Quantum Groups
- III.76 Quaternions, Octonions, and Normed Division Algebras
- III.77 Representations
- III.78 Ricci Flow
- III.79 Riemann Surfaces
- III.80 The Riemann Zeta Function
- III.81 Rings, Ideals, and Modules
- III.82 Schemes
- III.83 The Schrödinger Equation
- III.84 The Simplex Algorithm
- III.85 Special Functions
- III.86 The Spectrum
- III.87 Spherical Harmonics
- III.88 Symplectic Manifolds
- III.89 Tensor Products
- III.90 Topological Spaces
- III.91 Transforms
- III.92 Trigonometric Functions
- III.93 Universal Covers
- III.94 Variational Methods
- III.95 Varieties
- III.96 Vector Bundles
- III.97 Von Neumann Algebras
- III.98 Wavelets
- III.99 The Zermelo–Fraenkel Axioms
- Part IV Branches of Mathematics
- IV.1 Algebraic Numbers
- IV.2 Analytic Number Theory
- IV.3 Computational Number Theory
- IV.4 Algebraic Geometry
- IV.5 Arithmetic Geometry
- IV.6 Algebraic Topology
- IV.7 Differential Topology
- IV.8 Moduli Spaces
- IV.9 Representation Theory
- IV.10 Geometric and Combinatorial Group Theory
- IV.11 Harmonic Analysis
- IV.12 Partial Differential Equations
- IV.13 General Relativity and the Einstein Equations
- IV.14 Dynamics
- IV.15 Operator Algebras
- IV.16 Mirror Symmetry
- IV.17 Vertex Operator Algebras
- IV.18 Enumerative and Algebraic Combinatorics
- IV.19 Extremal and Probabilistic Combinatorics
- IV.20 Computational Complexity
- IV.21 Numerical Analysis
- IV.22 Set Theory
- IV.23 Logic and Model Theory
- IV.24 Stochastic Processes
- IV.25 Probabilistic Models of Critical Phenomena
- IV.26 High-Dimensional Geometry and Its Probabilistic Analogues
- Part V Theorems and Problems
- V.1 The ABC Conjecture
- V.2 The Atiyah–Singer Index Theorem
- V.3 The Banach–Tarski Paradox
- V.4 The Birch–Swinnerton-Dyer Conjecture
- V.5 Carleson’s Theorem
- V.6 The Central Limit Theorem
- V.7 The Classification of Finite Simple Groups
- V.8 Dirichiet’s Theorem
- V.9 Ergodic Theorems
- V.10 Fermat’s Last Theorem
- V.11 Fixed Point Theorems
- V.12 The Four-Color Theorem
- V.13 The Fundamental Theorem of Algebra
- V.14 The Fundamental Theorem of Arithmetic
- V.15 Gödel’s Theorem
- V.16 Gromov’s Polynomial-Growth Theorem
- V.17 Hilbert’s Nullstellensatz
- V.18 The Independence of the Continuum Hypothesis
- V.19 Inequalities
- V.20 The Insolubility of the Halting Problem
- V.21 The Insolubility of the Quintic
- V.22 Liouvflle’s Theorem and Roth’s Theorem
- V.23 Mostow’s Strong Rigidity Theorem
- V.24 The P versus NP Problem
- V.25 The Poincaré Conjecture
- V.26 The Prime Number Theorem and the Riemann Hypothesis
- V.27 Problems and Results in Additive Number Theory
- V.28 From Quadratic Reciprocity to Class Field Theory
- V.29 Rational Points on Curves and the Mordell Conjecture
- V.30 The Resolution of Singularities
- V.31 The Riemann–Roch Theorem
- V.32 The Robertson–Seymour Theorem
- V.33 The Three-Body Problem
- V.34 The Uniformization Theorem
- V.35 The Weil Conjectures
- Part VI Mathematicians
- VI.1 Pythagoras (ca. 569 B.C.E.–ca.494 B.C.E.)
- VI.2 Euclid (ca. 325 B.C.E.–ca. 265 B.C.E.)
- VI.3 Archimedes (ca. 287 B.C.E.–212 B.C.E.)
- VI.4 Apollonius (ca. 262 B.C.E.–ca. 190 B.C.E.)
- VI.5 Abu Ja’far Muhammad ibn Mūsā al-Khwārizmī (800–847)
- VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170–ca. 1250)
- VI.7 Girolamo Cardano (1501–1576)
- VI.8 Rafael Bombelli (1526-after 1572)
- VI.9 François Viète (1540–1603)
- VI.10 Simon Stevin (1548–1620)
- VI.11 René Descartes (1596–1650)
- VI.12 Pierre Fermat (160?-1665)
- VI.13 Blaise Pascal (1623–1662)
- VI.14 Isaac Newton (1642–1727)
- VI.15 Gottfried Wilhelm Leibniz (1646–1716)
- VI.16 Brook Taylor (1685–1731)
- VI.17 Christian Goldbach (1690–1764)
- VI.18 The Bernoullis (f1.18th century)
- VI.19 Leonhard Euler (1707–1783)
- VI.20 Jean Le Rond d’Alembert (1717–1783)
- VI.21 Edward Waring (ca.1735–1798)
- VI.22 Joseph Louis Lagrange (1736–1813)
- VI.23 Pierre-Simon Laplace (1749–1827)
- VI.24 Adrien-Marie Legendre (1752–1833)
- VI.25 Jean-Baptiste Joseph Fourier (1768–1830)
- VI.26 Carl Friedrich Gauss (1777–1855)
- VI.27 Siméon-Denis Poisson (1781–1840)
- VI.28 Bernard Bolzano (1781–1848)
- VI.29 Augustin-Louis Cauchy (1789–1857)
- VI.30 August Ferdinand Möbius (1790–1868)
- VI.31 Nicolai Ivanovich Lobachevskii (1792–1856)
- VI.32 George Green (1793–1841)
- VI.33 Niels Henrik Abel (1802–1829)
- VI.34 János Bolyai (1802–1860)
- VI.35 Carl Gustav Jacob Jacobi (1804–1851)
- VI.36 Peter Gustav Lejeune Dirichlet (1805–1859)
- VI.37 William Rowan Hamilton (1805–1865)
- VI.38 Augustus De Morgan (1806–1871)
- VI.39 Joseph Liouville (1809–1882)
- VI.40 Ernst Eduard Kummer (1810–1893)
- VI.41 Évariste Galois (1811–1832)
- VI.42 James Joseph Sylvester (1814–1897)
- VI.43 George Boole (1815–1864)
- VI.44 Karl Weierstrass (1815–1897)
- VI.45 Pafnuty Chebyshev (1821–1894)
- VI.46 Arthur Cayley (1821–1895)
- VI.47 Charles Hermite (1822–1901)
- VI.48 Leopold Kronecker (1823–1891)
- VI.49 Georg Friedrich Bernhard Riemann (1826–1866)
- VI.50 Julius Wilhelm Richard Dedekind (1831–1916)
- VI.51 Émile Léonard Mathieu (1835–1890)
- VI.52 Camille Jordan (1838–1922)
- VI.53 Sophus Lie (1842–1899)
- VI.54 Georg Cantor (1845–1918)
- VI.55 William Kingdon Clifford (1845–1879)
- VI.56 Gottlob Frege (1848–1925)
- VI.57 Christian Felix Klein (1849–1925)
- VI.58 Ferdinand Georg Frobenius (1849–1917)
- VI.59 Sofya (Sonya) Kovalevskaya (1850–1891)
- VI.60 William Burnside (1852–1927)
- VI.61 Jules Henri Poincaré (1854–1912)
- VI.62 Giuseppe Peano (1858–1932)
- VI.63 David Hilbert (1862–1943)
- VI.64 Hermann Minkowski (1864–1909)
- VI.65 Jacques Hadamard (1865–1963)
- VI.66 Ivar Fredholm (1866–1927)
- VI.67 Charles-Jean de la Vallée Poussin (1866–1962)
- VI.68 Felix Hausdorff (1868–1942)
- VI.69 Élie Joseph Cartan (1869–1951)
- VI.70 Emile Borel (1871–1956)
- VI.71 Bertrand Arthur William Russell (1872–1970)
- VI.72 Henri Lebesgue (1875–1941)
- VI.73 Godfrey Harold Hardy (1877–1947)
- VI.74 Frigyes (Frédéric) Riesz (1880–1956)
- VI.75 Luitzen Egbertus Jan Brouwer (1881–1966)
- VI.76 Emmy Noether (1882–1935)
- VI.77 Waclaw Sierpiński (1882–1969)
- VI.78 George Birkhoff (1884–1944)
- VI.79 John Edensor Littlewood (1885–1977)
- VI.80 Hermann Weyl (1885–1955)
- VI.81 Thoralf Skolem (1887–1963)
- VI.82 Srinwasa Ramanujan (1887–1920)
- VI.83 Richard Courant (1888–1972)
- VI.84 Stefan Banach (1892–1945)
- VI.85 Norbert Wiener (1894–1964)
- VI.86 Emil Artin (1898–1962)
- VI.87 Alfred Tarski (1901–1983)
- VI.88 Andrei Nikolaevich Kolmogorov (1903–1987)
- VI.89 Alonzo Church (1903–1995)
- VI.90 William Valiance Douglas Hodge (1903–1975)
- VI.91 John von Neumann (1903–1957)
- VI.92 Kurt Gödel (1906–1978)
- VI.93 André Weil (1906–1998)
- VI.94 Alan Turing (1912–1954)
- VI.95 Abraham Robinson (1918–1974)
- VI.96 Nicolas Bourbaki (1935–)
- Part VII The Influence of Mathematics
- VII.1 Mathematics and Chemistry
- VII.2 Mathematical Biology
- VII.3 Wavelets and Applications
- VII.4 The Mathematics of Traffic in Networks
- VII.5 The Mathematics of Algorithm Design
- VII.6 Reliable Transmission of Information
- VII.7 Mathematics and Cryptography
- VII.8 Mathematics and Economic Reasoning
- VII.9 The Mathematics of Money
- VII.10 Mathematical Statistics
- VII.11 Mathematics and Medical Statistics
- VII.12 Analysis, Mathematical and Philosophical
- VII.13 Mathematics and Music
- VII.14 Mathematics and Art
- Part VIII Final Perspectives
- Index
V.6 The Central Limit Theorem
The central limit theorem is a fundamental result in probability concerning sums of independent random variables. Let X1, X2, . . . be independent and suppose that they are identically distributed. Suppose also that they have mean 0 and variance 1. Then X1 + · · · + Xn has mean 0 and variance n. (The variance is n because the Xi are independent.) Therefore, Yn = (X1 + · · · + Xn) / 
has mean 0 and variance 1. The central limit theorem states that, regardless of the distribution of the Xi, the random variable Yn converges to a standard normal distribution. It is easy to deduce from this a similar result for random variables with any finite mean and variance. Details may be found in PROBABILITY DISTRIBUTIONS [III.71 §5].
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