 Outline of probability

Probability is the likelihood or chance that something is the case or will happen. 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.
The following outline is provided as an overview and guide to the variety of topics included within the subject of probability.
Contents
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
Conditional probability
 The law of total probability
 Likelihood
 Bayes' theorem
 Bayesian probability
Independence
Probability theory
(Related topics: measure theory)
Measuretheoretic probability
Independence
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
 General moments about the mean
 Correlated and uncorrelated random variables
 Conditional expectation:
 Fatou's lemma and the monotone and dominated convergence theorems
 Markov's inequality and Chebyshev's inequality
Independence
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,
 χsquared (or chisquared),
 Fdistribution,
 Student's tdistribution, and
 Cauchy.
Some other distributions
 Cantor
 FisherTippett (or Gumbel)
 Pareto
 Benford's law
Functions of random variables
 Sums of random variables*
 General functions of random variables*
 Borel's paradox
Generating functions
(Related topics: integral transforms)
Common generating functions
 Probabilitygenerating functions
 Momentgenerating functions
 Laplace transforms and LaplaceStieltjes transforms
 Characteristic functions
Applications
 A proof of the central limit theorem*
 Random sums of random variables*
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
 BerryEsséen 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
 OrnsteinUhlenbeck process
 Gamma process
Markov processes
 Markov property
 Branching process
 Markov chain
 Population processes
 Applications to queueing theory
Stochastic differential equations
Time series
 Movingaverage* and autoregressive* processes
 Correlation function and autocorrelation
Martingales
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
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