This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates a distribution with two random variables, discrete or continuous. Other codes are just abbreviations for topics. The list of codes can be found in the table of contents.
Core probability: selected topics
Probability theory
Basic notions (bsc)
Random variable Continuous probability distribution / (1:C) Cumulative distribution function / (1:DCR) Discrete probability distribution / (1:D) Independent and identically-distributed random variables / (FS:BDCR) Joint probability distribution / (F:DC) Marginal distribution / (2F:DC) Probability density function / (1:C) Probability distribution / (1:DCRG) Probability distribution function Probability mass function / (1:D) Sample space
Instructive examples (paradoxes) (iex)
Berkson's paradox / (2:B) Bertrand's box paradox / (F:B) Borel–Kolmogorov paradox / cnd (2:CM) Boy or Girl paradox / (2:B) Exchange paradox / (2:D) Intransitive dice Monty Hall problem / (F:B) Necktie paradox Simpson's paradox Sleeping Beauty problem St. Petersburg paradox / mnt (1:D) Three Prisoners problem Two envelopes problem
Moments (mnt)
Expected value / (12:DCR) Canonical correlation / (F:R) Carleman's condition / anl (1:R) Central moment / (1:R) Coefficient of variation / (1:R) Correlation / (2:R) Correlation function / (U:R) Covariance / (2F:R) (1:G) Covariance function / (U:R) Covariance matrix / (F:R) Cumulant / (12F:DCR) Factorial moment / (1:R) Factorial moment generating function / anl (1:R) Fano factor Geometric standard deviation / (1:R) Hamburger moment problem / anl (1:R) Hausdorff moment problem / anl (1:R) Isserlis Gaussian moment theorem / Gau Jensen's inequality / (1:DCR) Kurtosis / (1:CR) Law of the unconscious statistician / (1:DCR) Moment / (12FU:CRG) Law of total covariance / (F:R) Law of total cumulance / (F:R) Law of total expectation / (F:DR) Law of total variance / (F:R) Logmoment generating function Marcinkiewicz–Zygmund inequality / inq Method of moments / lmt (L:R) Moment problem / anl (1:R) Moment-generating function / anl (1F:R) Second moment method / (1FL:DR) Skewness / (1:R) St. Petersburg paradox / iex (1:D) Standard deviation / (1:DCR) Standardized moment / (1:R) Stieltjes moment problem / anl (1:R) Trigonometric moment problem / anl (1:R) Uncorrelated / (2:R) Variance / (12F:DCR) Variance-to-mean ratio / (1:R)
Inequalities (inq)
Chebyshev's inequality / (1:R) An inequality on location and scale parameters / (1:R) Azuma's inequality / (F:BR) Bennett's inequality / (F:R) Bernstein inequalities / (F:R) Bhatia–Davis inequality Chernoff bound / (F:B) Doob's martingale inequality / (FU:R) Dudley's theorem / Gau Entropy power inequality Etemadi's inequality / (F:R) Gauss's inequality Hoeffding's inequality / (F:R) Khintchine inequality / (F:B) Kolmogorov's inequality / (F:R) Marcinkiewicz–Zygmund inequality / mnt Markov's inequality / (1:R) McDiarmid's inequality Multidimensional Chebyshev's inequality Paley–Zygmund inequality / (1:R) Pinsker's inequality / (2:R) Vysochanskiï–Petunin inequality / (1:C)
Markov chains, processes, fields, networks (Mar)
Markov chain / (FLSU:D) Additive Markov chain Bayesian network / Bay Birth–death process / (U:D) CIR process / scl Chapman–Kolmogorov equation / (F:DC) Cheeger bound / (L:D) Conductance Contact process Continuous-time Markov process / (U:D) Detailed balance / (F:D) Examples of Markov chains / (FL:D) Feller process / (U:G) Fokker–Planck equation / scl anl Foster's theorem / (L:D) Gauss–Markov process / Gau Geometric Brownian motion / scl Hammersley–Clifford theorem / (F:C) Harris chain / (L:DC) Hidden Markov model / (F:D) Hidden Markov random field Hunt process / (U:R) Kalman filter / (F:C) Kolmogorov backward equation / scl Kolmogorov's criterion / (F:D) Kolmogorov's generalized criterion / (U:D) Krylov–Bogolyubov theorem / anl Lumpability Markov additive process Markov blanket / Bay Markov chain mixing time / (L:D) Markov decision process Markov information source Markov kernel Markov logic network Markov network Markov process / (U:D) Markov property / (F:D) Markov random field Master equation / phs (U:D) Milstein method / scl Moran process Ornstein–Uhlenbeck process / Gau scl Partially observable Markov decision process Product-form solution / spr Quantum Markov chain / phs Semi-Markov process Stochastic matrix / anl Telegraph process / (U:B) Variable-order Markov model Wiener process / Gau scl
Gaussian random variables, vectors, functions (Gau)
Normal distribution / spd Abstract Wiener space Brownian bridge Classical Wiener space Concentration dimension Dudley's theorem / inq Estimation of covariance matrices Fractional Brownian motion Gaussian isoperimetric inequality Gaussian measure / anl Gaussian random field Gauss–Markov process / Mar Integration of the normal density function / spd anl Gaussian process Isserlis Gaussian moment theorem / mnt Karhunen–Loève theorem Large deviations of Gaussian random functions / lrd Lévy's modulus of continuity theorem / (U:R) Matrix normal distribution / spd Multivariate normal distribution / spd Ornstein–Uhlenbeck process / Mar scl Paley–Wiener integral / anl Pregaussian class Schilder's theorem / lrd Wiener process / Mar scl
Conditioning (cnd)
Conditioning / (2:BDCR) Bayes' theorem / (2:BCG) Borel–Kolmogorov paradox / iex (2:CM) Conditional expectation / (2:BDR) Conditional independence / (3F:BR) Conditional probability Conditional probability distribution / (2:DC) Conditional random field / (F:R) Disintegration theorem / anl (2:G) Inverse probability / Bay Luce's choice axiom Regular conditional probability / (2:G) Rule of succession / (F:B)
Specific distributions (spd)
Binomial distribution / (1:D) (a,b,0) class of distributions / (1:D) Anscombe transform Bernoulli distribution / (1:B) Beta distribution / (1:C) Bose–Einstein statistics / (F:D) Cantor distribution / (1:C) Cauchy distribution / (1:C) Chi-squared distribution / (1:C) Compound Poisson distribution / (F:DR) Degenerate distribution / (1:D) Dirichlet distribution / (F:C) Discrete phase-type distribution / (1:D) Erlang distribution / (1:C) Exponential-logarithmic distribution / (1:C) Exponential distribution / (1:C) F-distribution / (1:C) Fermi–Dirac statistics / (1F:D) Fisher–Tippett distribution / (1:C) Gamma distribution / (1:C) Generalized normal distribution / (1:C) Geometric distribution / (1:D) Half circle distribution / (1:C) Hypergeometric distribution / (1:D) Normal distribution / Gau Integration of the normal density function / Gau anl Lévy distribution / (1:C) Matrix normal distribution / Gau Maxwell–Boltzmann statistics / (F:D) McCullagh's parametrization of the Cauchy distributions / (1:C) Multinomial distribution / (F:D) Multivariate normal distribution / Gau Negative binomial distribution / (1:D) Pareto distribution / (1:C) Phase-type distribution / (1:C) Poisson distribution / (1:D) Power law / (1:C) Skew normal distribution / (1:C) Stable distribution / (1:C) Student's t-distribution / (1:C) Tracy–Widom distribution / rmt Triangular distribution / (1:C) Weibull distribution / (1:C) Wigner semicircle distribution / (1:C) Wishart distribution / (F:C) Zeta distribution / (1:D) Zipf's law / (1:D)
Empirical measure (emm)
Donsker's theorem / (LU:C) Empirical distribution function Empirical measure / (FL:RG) (U:D) Empirical process / (FL:RG) (U:D) Glivenko–Cantelli theorem / (FL:RG) (U:D) Khmaladze transformation / (FL:RG) (U:D) Vapnik–Chervonenkis theory
Limit theorems (lmt)
Central limit theorem / (L:R) Berry–Esseen theorem / (F:R) Characteristic function / anl (1F:DCR) De Moivre–Laplace theorem / (L:BD) Helly–Bray theorem / anl (L:R) Illustration of the central limit theorem / (L:DC) Lindeberg's condition Lyapunov's central limit theorem / (L:R) Lévy's continuity theorem / anl (L:R) Lévy's convergence theorem / (S:R) Martingale central limit theorem / (S:R) Method of moments / mnt (L:R) Slutsky's theorem / anl Weak convergence of measures / anl
Large deviations (lrd)
Large deviations theory Contraction principle Cramér's theorem Exponentially equivalent measures Freidlin–Wentzell theorem Laplace principle Large deviations of Gaussian random functions / Gau Rate function Schilder's theorem / Gau Tilted large deviation principle Varadhan's lemma
Random graphs (rgr)
Random graph BA model Barabási–Albert model Erdős–Rényi model Percolation theory / phs (L:B) Percolation threshold / phs Random geometric graph Random regular graph Watts and Strogatz model
Random matrices (rmt)
Random matrix Circular ensemble Gaussian matrix ensemble Tracy–Widom distribution / spd Weingarten function / anl
Stochastic calculus (scl)
Itô calculus Bessel process CIR process / Mar Doléans-Dade exponential Dynkin's formula Euler–Maruyama method Feynman–Kac formula Filtering problem Fokker–Planck equation / Mar anl Geometric Brownian motion / Mar Girsanov theorem Green measure Heston model / fnc Hörmander's condition / anl Infinitesimal generator Itô's lemma Itô calculus Itô diffusion Itô isometry Itô's lemma Kolmogorov backward equation / Mar Local time Milstein method / Mar Novikov's condition Ornstein–Uhlenbeck process / Gau Mar Quadratic variation Random dynamical system / rds Reversible diffusion Runge–Kutta method Russo–Vallois integral Schramm–Loewner evolution Semimartingale Stochastic calculus Stochastic differential equation Stochastic processes and boundary value problems / anl Stratonovich integral Tanaka equation Tanaka's formula Wiener process / Gau Mar Wiener sausage
Malliavin calculus (Mal)
Malliavin calculus Clark–Ocone theorem H-derivative Integral representation theorem for classical Wiener space Integration by parts operator Malliavin derivative Malliavin's absolute continuity lemma Ornstein–Uhlenbeck operator Skorokhod integral
Random dynamical systems (rds) Random dynamical system / scl Absorbing set Base flow Pullback attractor
Analytic aspects (including measure theoretic) (anl)
Probability space Carleman's condition / mnt (1:R) Characteristic function / lmt (1F:DCR) Contiguity#Probability theory Càdlàg Disintegration theorem / cnd (2:G) Dynkin system Exponential family Factorial moment generating function / mnt (1:R) Filtration Fokker–Planck equation / scl Mar Gaussian measure / Gau Hamburger moment problem / mnt (1:R) Hausdorff moment problem / mnt (1:R) Helly–Bray theorem / lmt (L:R) Hörmander's condition / scl Integration of the normal density function / spd Gau Kolmogorov extension theorem / (SU:R) Krylov–Bogolyubov theorem / Mar Law (stochastic processes) / (U:G) Location-scale family Lévy's continuity theorem / lmt (L:R) Minlos' theorem Moment problem / mnt (1:R) Moment-generating function / mnt (1F:R) Natural filtration / (U:G) Paley–Wiener integral / Gau Sazonov's theorem Slutsky's theorem / lmt Standard probability space Stieltjes moment problem / mnt (1:R) Stochastic matrix / Mar Stochastic processes and boundary value problems / scl Trigonometric moment problem / mnt (1:R) Weak convergence of measures / lmt Weingarten function / rmt
Core probability: other articles, by number and type of random variables A single random variable (1:) Binary (1:B)
Bernoulli trial / (1:B) Complementary event / (1:B) Entropy / (1:BDC) Event / (1:B) Indecomposable distribution / (1:BDCR) Indicator function / (1F:B)
Discrete (1:D)
Binomial probability / (1:D)