The following outline is provided as an overview and guide to probability:
Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are 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 are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty), we call 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 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 (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
Functions of random variables
Sums of random variables
General functions of 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
Random sums of random variables
Convergence of random variables
(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