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