Editing Outline of probability

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{{Short description|1=Overview of and topical guide to probability}}
{{Probability topics sidebar}}
[[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.

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==Introduction==
* [[Probability]] and [[randomness]].

==Basic probability==
(Related topics: [[set theory]], [[simple theorems in the algebra of sets]])

===Events===
* [[Event (probability theory)|Events in probability theory]]
* [[Elementary event]]s, [[sample space]]s, [[Venn diagram]]s
* [[Mutually exclusive|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 space]]s, [[sigma-algebra|σ-algebras]] and [[probability measure]]s
* [[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 lemma]]s 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 variable]]s==
===Discrete and continuous random variables===
* [[Discrete random variable]]s: [[Probability mass function]]s
* [[Continuous random variable]]s: [[Probability density function]]s
* [[Normalizing constant]]s
* [[Cumulative distribution function]]s
* [[Joint distribution|Joint]], [[marginal distribution|marginal]] and [[Conditional distribution|conditional]] distributions

===Expectation===
* [[expected value|Expectation]] (or [[mean]]), [[variance]] and [[covariance]]
** [[Jensen's inequality]]
* General [[Moment about the mean|moments about the mean]]
* [[Correlation|Correlated]] and [[uncorrelated]] random variables
* [[Conditional expectation]]:
** [[law of total expectation]], [[law of total variance]]
* [[Fatou's lemma]] and the [[monotone convergence theorem|monotone]] and [[dominated convergence theorem|dominated]] convergence theorems
* [[Markov's inequality]] and [[Chebyshev's inequality]]

===Independence===
* [[Independent random variables]]

===Some common distributions===
* Discrete:
** [[Constant random variable|constant]] (see also [[degenerate distribution]]),
** [[Bernoulli distribution|Bernoulli]] and [[Binomial distribution|binomial]],
** [[Negative binomial distribution|negative binomial]],
** [[Uniform distribution (discrete)|(discrete) uniform]],
** [[geometric distribution|geometric]],
** [[Poisson distribution|Poisson]], and
** [[hypergeometric distribution|hypergeometric]].
* Continuous:
** [[Uniform distribution (continuous)|(continuous) uniform]],
** [[exponential distribution|exponential]],
** [[gamma distribution|gamma]],
** [[beta distribution|beta]],
** [[normal distribution|normal]] (or ''Gaussian'') and [[multivariate normal distribution|multivariate normal]],
** [[chi-squared distribution|χ-squared]] (or chi-squared),
** [[F-distribution]],
** [[Student's t-distribution]], and
** [[Cauchy distribution|Cauchy]].

===Some other distributions===
* [[Cantor distribution|Cantor]]
* [[Fisher–Tippett distribution|Fisher–Tippett]] (or ''Gumbel'')
* [[Pareto distribution|Pareto]]
* [[Benford's law]]

===Functions of random variables===
* [[Sum of normally distributed random variables]]
* [[Borel's paradox]]

==Generating functions==
(Related topics: [[integral transform]]s)

===Common generating functions===
* [[Probability-generating function]]s
* [[Moment-generating function]]s
* [[Laplace transform]]s and [[Laplace–Stieltjes transform]]s
* [[Characteristic function (probability theory)|Characteristic function]]s

===Applications===
* [[Central limit theorem#Proof|A proof of the central limit theorem]]

==Convergence of random variables==
(Related topics: [[Modes of convergence (annotated index)|convergence]])

===Modes of convergence===
* [[Convergence in distribution]] and [[convergence in probability]],
* Convergence in [[Convergence of random variables#Convergence in mean|mean]], [[Convergence of random variables#Convergence in mean square|mean square]] and [[Convergence of random variables#Convergence in rth mean|''r''th mean]]
* [[Almost sure convergence]]
* [[Skorokhod's representation theorem]]

===Applications===
* [[Central limit theorem]] and [[Law of large numbers|Laws of large numbers]]
** [[Illustration of the central limit theorem]] and a [[concrete illustration of the central limit theorem|'concrete' illustration]]
** [[Berry–Esseen theorem|Berry–Esséen theorem]]
* [[Law of the iterated logarithm]]

==[[Stochastic process]]es==
===Some common [[stochastic process]]es===
* [[Random walk]]
* [[Poisson process]]
* [[Compound Poisson process]]
* [[Wiener process]]
* [[Geometric Brownian motion]]
* [[Fractional Brownian motion]]
* [[Brownian bridge]]
* [[Ornstein–Uhlenbeck process]]
* [[Gamma process]]

===Markov processes===
* [[Markov property]]
* [[Branching process]]
** [[Galton–Watson process]]
* [[Markov chain]]
** [[Examples of Markov chains]]
* [[Population process]]es
* Applications to [[queueing theory]]
** [[Erlang distribution]]

===Stochastic differential equations===
* [[Stochastic calculus]]
* [[Diffusion]]s
** [[Brownian motion]]
** [[Wiener equation]]
** [[Wiener process]]

===[[Time series]]===
* [[Moving average model|Moving-average]] and [[autoregressive process|autoregressive]] processes
* [[Correlation function]] and [[autocorrelation]]

===[[Martingale (probability theory)|Martingales]]===
* [[Martingale central limit theorem]]
* [[Azuma's inequality]]

==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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