Quantity[ edit]
Quantity –
• Arithmetic –
• Natural numbers –
• Integers –
• Rational numbers –
• Real numbers –
• Complex numbers –
• Hypercomplex numbers –
• Infinity –

Matrices[ edit]
• 2 × 2 real matrices
• Matrix theory
• Matrix addition
• Matrix multiplication
• Basis transformation matrix
• Characteristic polynomial
• Trace
• Eigenvalue, eigenvector and eigenspace
• Cayley–Hamilton theorem
• Spread of a matrix
• Jordan normal form
• Weyr canonical form
• Rank
• Matrix inversion, invertible matrix
• Pseudoinverse
• Adjugate
• Transpose
• Dot product
• Symmetric matrix
• Orthogonal matrix
• Skew-symmetric matrix
• Conjugate transpose
• Unitary matrix
• Hermitian matrix, Antihermitian matrix
• Positive-definite, positive-semidefinite matrix
• Pfaffian
• Projection
• Spectral theorem
• Perron–Frobenius theorem
• List of matrices
• Diagonal matrix, main diagonal
• Diagonalizable matrix
• Triangular matrix
• Tridiagonal matrix
• Block matrix
• Sparse matrix
• Hessenberg matrix
• Hessian matrix
• Vandermonde matrix
• Stochastic matrix
• Toeplitz matrix
• Circulant matrix
• Hankel matrix
• (0,1)-matrix

Relations[ edit]
• Matrix equivalence
• Matrix congruence
• Matrix similarity
• Matrix consimilarity
• Row equivalence

Vector spaces[ edit]
• Linear combination
• Linear span
• Linear independence
• Scalar multiplication
• Basis
• Change of basis
• Hamel basis
• Cyclic decomposition theorem
• Dimension theorem for vector spaces
• Hamel dimension
• Examples of vector spaces
• Linear map
• Shear mapping or Galilean transformation
• Squeeze mapping or Lorentz transformation
• Linear subspace
• Row and column spaces
• Column space
• Row space
• Cyclic subspace
• Null space, nullity
• Rank-nullity theorem
• Nullity theorem
• Dual space
• Linear function
• Linear functional
• Category of vector spaces

• Number theory –

Space[ edit]
Space –
• Geometry –
• Algebraic geometry –
• Trigonometry –
• Differential geometry –
• Topology –
• Fractal geometry –

• List of calculus topics

Differential calculus[ edit]
• Derivative
• Notation
• Newton's notation for differentiation
• Leibniz's notation for differentiation
• Simplest rules
• Derivative of a constant
• Sum rule in differentiation
• Constant factor rule in differentiation
• Linearity of differentiation
• Power rule
• Derivative (examples)
• Chain rule
• local linearization
• Product rule
• Quotient rule
• Inverse functions and differentiation
• Implicit differentiation
• Stationary point
• Maxima and minima
• First derivative test
• Second derivative test
• Extreme value theorem
• Differential equation
• Differential operator
• Newton's method
• Taylor's theorem
• L'Hôpital's rule
• General Leibniz rule
• Mean value theorem
• Logarithmic derivative
• Differential (calculus)
• Related rates
• Regiomontanus' angle maximization problem

Special functions and numbers[ edit]
• Natural logarithm
• e (mathematical constant)
• Exponential function
• Hyperbolic angle
• Hyperbolic function
• Stirling's approximation
• Bernoulli numbers

Multivariable[ edit]
• Partial derivative
• Disk integration
• Shell integration
• Gabriel's horn
• Jacobian matrix
• Hessian matrix
• Curvature
• Green's theorem
• Divergence theorem
• Stokes' theorem

Applied mathematics[ edit]
Applied mathematics –
• Mathematical physics –
• Analytical mechanics –
• Mathematical fluid dynamics –
• Numerical analysis –
• Mathematical optimization –
• Statistics –
• Mathematical economics –
• Financial mathematics –
• Game theory –
• Mathematical biology –
• Cryptography –
• Operations research –
• Information theory –
• Control theory –
• Dynamical systems –

General aspects[ edit]
• Probability
• Randomness, Pseudorandomness, Quasirandomness
• Randomization, hardware random number generator
• Random number generator
• Random sequence
• Uncertainty
• Statistical dispersion
• Observational error
• Equiprobable
• Equipossible
• Average
• Probability interpretations
• Markovian
• Statistical regularity
• Central tendency
• Bean machine
• Relative frequency
• Frequency probability
• Maximum likelihood
• Bayesian probability
• Principle of indifference
• Credal set
• Cox's theorem
• Principle of maximum entropy
• Information entropy
• Urn problems
• Extractor
• Aleatoric, aleatoric music
• Free probability
• Exotic probability
• Schrödinger method
• Empirical measure
• Glivenko–Cantelli theorem
• Zero-one law
• Kolmogorov's zero-one law
• Hewitt–Savage zero-one law
• Law of Truly Large Numbers
• Littlewood's law
• Infinite monkey theorem
• Littlewood–Offord problem
• Inclusion-exclusion principle
• Impossible event
• Information geometry
• Talagrand's concentration inequality

Random variables[ edit]
• Discrete random variable
• Probability mass function
• Constant random variable
• Expected value
• Jensen's inequality
• Variance
• Standard deviation
• Geometric standard deviation
• Multivariate random variable
• Joint probability distribution
• Marginal distribution
• Kirkwood approximation
• Independent identically-distributed random variables
• Independent and identically-distributed random variables
• Statistical independence
• Conditional independence
• Pairwise independence
• Covariance
• Covariance matrix
• De Finetti's theorem
• Correlation
• Uncorrelated
• Correlation function
• Canonical correlation
• Convergence of random variables
• Weak convergence of measures
• Helly–Bray theorem
• Slutsky's theorem
• Skorokhod's representation theorem
• Lévy's continuity theorem
• Uniform integrability
• Markov's inequality
• Chebyshev's inequality = Chernoff bound
• Chernoff's inequality
• Bernstein inequalities (probability theory)
• Hoeffding's inequality
• Kolmogorov's inequality
• Etemadi's inequality
• Khintchine inequality
• Paley–Zygmund inequality
• Laws of large numbers
• Asymptotic equipartition property
• Typical set
• Law of large numbers
• Kolmogorov's two-series theorem
• Random field
• Conditional random field
• Borel–Cantelli lemma
• Wick product

Theory of probability distributions[ edit]
• Probability distribution
• Probability distribution function
• Probability density function
• Probability mass function
• Cumulative distribution function
• Quantile
• Moment (mathematics)
• Moment about the mean
• Standardized moment
• Skewness
• Kurtosis
• Locality
• Cumulant
• Factorial moment
• Expected value
• Law of the unconscious statistician
• Second moment method
• Variance
• Coefficient of variation
• Variance-to-mean ratio
• Covariance function
• An inequality on location and scale parameters
• Taylor expansions for the moments of functions of random variables
• Moment problem
• Hamburger moment problem
• Carleman's condition
• Hausdorff moment problem
• Trigonometric moment problem
• Stieltjes moment problem
• Prior probability distribution
• Total variation distance
• Hellinger distance
• Wasserstein metric
• Lévy–Prokhorov metric
• Lévy metric
• Continuity correction
• Heavy-tailed distribution
• Truncated distribution
• Infinite divisibility
• Stability (probability)
• Indecomposable distribution
• Power law
• Anderson's theorem
• Probability bounds analysis
• Probability box

Algorithmics[ edit]
• Probable prime
• Probabilistic algorithm = Randomised algorithm
• Monte Carlo method
• Las Vegas algorithm
• Probabilistic Turing machine
• Stochastic programming
• Probabilistically checkable proof
• Box–Muller transform
• Metropolis algorithm
• Gibbs sampling
• Inverse transform sampling method
• Walk-on-spheres method

Measure-theoretic probability[ edit]
• Sample spaces, σ-algebras and probability measures
• Probability space
• Sample space
• Standard probability space
• Random element
• Random compact set
• Dynkin system
• Probability axioms
• Event (probability theory)
• Complementary event
• Elementary event
• " Almost surely"
Independence[ edit]
• Independence (probability theory)
• The Borel–Cantelli lemmas and Kolmogorov's zero-one law
Conditional probability[ edit]
• Conditional probability
• Conditioning (probability)
• Conditional expectation
• Conditional probability distribution
• Regular conditional probability
• Disintegration theorem
• Bayes' theorem
• Rule of succession
• Conditional independence
• Conditional event algebra
• Goodman–Nguyen–van Fraassen algebra
Random variables[ edit]
Discrete and continuous random variables[ edit]
• Discrete random variables: Probability mass functions
• Continuous random variable: Probability density functions
• Normalizing constants
• Cumulative distribution functions
• Joint, marginal and conditional distributions
Expectation[ edit]
• Expectation (or mean), variance and covariance
• Jensen's inequality
• General moments about the mean
• Correlated and uncorrelated random variables
• Conditional expectation:
• law of total expectation, law of total variance
• Fatou's lemma and the monotone and dominated convergence theorems
• Markov's inequality and Chebyshev's inequality
Independence[ edit]
• Independent random variables
Some common distributions[ edit]
• Discrete:
• constant (see also degenerate distribution),
• Bernoulli and binomial,
• negative binomial,
• (discrete) uniform,
• geometric,
• Poisson, and
• hypergeometric.
• Continuous:
• (continuous) uniform,
• exponential,
• gamma,
• beta,
• normal (or Gaussian) and multivariate normal,
• χ-squared (or chi-squared),
• F-distribution,
• Student's t-distribution, and
• Cauchy.
Some other distributions[ edit]
• Cantor
• Fisher–Tippett (or Gumbel)
• Pareto
• Benford's law
Functions of random variables[ edit]
• Sums of random variables
• General functions of random variables
• Borel's paradox
Generating functions[ edit]
(Related topics: integral transforms)
Common generating functions[ edit]
• Probability-generating functions
• Moment-generating functions
• Laplace transforms and Laplace–Stieltjes transforms
• Characteristic functions
Applications[ edit]
• A proof of the central limit theorem
• Random sums of random variables