Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition whose truth is not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain is it that the event will occur?" The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
Introduction Probability and randomness.
Basic probability (Related topics: set theory, simple theorems in the algebra of sets)
Events Events in probability theory Elementary events, sample spaces, Venn diagrams Mutual exclusivity
Elementary probability The axioms of probability Boole's inequality
Meaning of probability Probability interpretations Bayesian probability Frequency probability
Calculating with probabilities Conditional probability The law of total probability Bayes' theorem
Independence Independence (probability theory)
Probability theory (Related topics: measure theory)
Measure-theoretic probability 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 Independence (probability theory) The Borel–Cantelli lemmas and Kolmogorov's zero–one law
Conditional probability 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
Discrete and continuous random variables Discrete random variables: Probability mass functions Continuous random variables: Probability density functions Normalizing constants Cumulative distribution functions Joint, marginal and conditional distributions
Expectation 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 Independent random variables
Some common distributions 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 Cantor Fisher–Tippett (or Gumbel) Pareto Benford's law
Functions of random variables Sum of normally distributed random variables Borel's paradox
Generating functions (Related topics: integral transforms)
Common generating functions Probability-generating functions Moment-generating functions Laplace transforms and Laplace–Stieltjes transforms Characteristic functions
Applications A proof of the central limit theorem
Convergence of random variables (Related topics: convergence)
Modes of convergence Convergence in distribution and convergence in probability, Convergence in mean, mean square and rth mean Almost sure convergence Skorokhod's representation theorem
Applications Central limit theorem and Laws of large numbers Illustration of the central limit theorem and a 'concrete' illustration Berry–Esséen theorem Law of the iterated logarithm
Stochastic processes
Some common stochastic processes Random walk Poisson process Compound Poisson process Wiener process Geometric Brownian motion Fractional Brownian motion Brownian bridge Ornstein–Uhlenbeck process Gamma process
Markov processes Markov property Branching process Galton–Watson process Markov chain Examples of Markov chains Population processes Applications to queueing theory Erlang distribution
Stochastic differential equations Stochastic calculus Diffusions Brownian motion Wiener equation Wiener process
Time series Moving-average and autoregressive processes Correlation function and autocorrelation
Martingales Martingale central limit theorem Azuma's inequality
See also Catalog of articles in probability theory Glossary of probability and statistics Notation in probability and statistics List of mathematical probabilists List of probability distributions List of probability topics List of scientific journals in probability Timeline of probability and statistics Topic outline of statistics