• Probability and randomness.

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

Events edit

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

Elementary probability edit

• The axioms of probability
• Boole's inequality

Meaning of probability edit

• Probability interpretations
• Bayesian probability
• Frequency probability

Calculating with probabilities edit

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

Independence edit

• Independence (probability theory)

(Related topics: measure theory)

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

Discrete and continuous random variables edit

• Discrete random variables: Probability mass functions
• Continuous random variables: 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

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

(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

(Related topics: convergence)

Modes of convergence edit

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

Applications edit

• 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

Some common stochastic processes edit

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

Markov processes edit

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

Stochastic differential equations edit

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

Time series edit

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

Martingales edit

• 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