Outline of probability

Probability
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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.

• Probability and randomness.

(Related topics: set theory, simple theorems in the algebra of sets)

• Events in probability theory
• Elementary events, sample spaces, Venn diagrams
• Mutual exclusivity

• The axioms of probability
• Boole's inequality

• Probability interpretations
• Bayesian probability
• Frequency probability

• Conditional probability
• The law of total probability
• Bayes' theorem

• Independence (probability theory)

(Related topics: measure theory)

• 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 (probability theory)
• The Borel–Cantelli lemmas and Kolmogorov's zero–one law

• 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

• Discrete random variables: Probability mass functions
• Continuous random variables: Probability density functions
• Normalizing constants
• Cumulative distribution functions
• Joint, marginal and conditional distributions

• 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

• Independent random variables

• 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.

• Cantor
• Fisher–Tippett (or Gumbel)
• Pareto
• Benford's law

• Sum of normally distributed random variables
• Borel's paradox

(Related topics: integral transforms)

• Probability-generating functions
• Moment-generating functions
• Laplace transforms and Laplace–Stieltjes transforms
• Characteristic functions

• A proof of the central limit theorem

(Related topics: convergence)

• Convergence in distribution and convergence in probability,
• Convergence in mean, mean square and rth mean
• Almost sure convergence
• Skorokhod's representation theorem

• 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

• Random walk
• Poisson process
• Compound Poisson process
• Wiener process
• Geometric Brownian motion
• Fractional Brownian motion
• Brownian bridge
• Ornstein–Uhlenbeck process
• Gamma process

• Markov property
• Branching process
  • Galton–Watson process
• Markov chain
  • Examples of Markov chains
• Population processes
• Applications to queueing theory
  • Erlang distribution

• Stochastic calculus
• Diffusions
  • Brownian motion
  • Wiener equation
  • Wiener process

• Moving-average and autoregressive processes
• Correlation function and autocorrelation

• Martingale central limit theorem
• Azuma's inequality

• 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