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
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
,
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
science and
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
Introduction
*
Probability and
randomness.
Basic probability
(Related topics:
set theory,
simple theorems in the algebra of sets The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement ( postfix ') of sets.
These properties assume the existence of ...
)
Events
*
Events in probability theory
*
Elementary events,
sample spaces,
Venn diagrams
*
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
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
Independence
*
Independence (probability theory)
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of ...
Probability theory
(Related topics:
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
)
Measure-theoretic probability
*
Sample spaces,
σ-algebras and
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
s
*
Probability space
**
Sample space
**
Standard probability space
In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin i ...
**
Random element
***
Random compact set
**
Dynkin system
*
Probability axioms
*
Event (probability theory)
**
Complementary event
*
Elementary event
* "
Almost surely"
Independence
*
Independence (probability theory)
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of ...
* The
Borel–Cantelli lemma
In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first ...
s and
Kolmogorov's zero–one law
Conditional probability
*
Conditional probability
*
Conditioning (probability)
*
Conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
*
Conditional probability distribution
*
Regular conditional probability In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution is a parametrized family of probability measures c ...
*
Disintegration theorem
In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related ...
*
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
*
Rule of succession
*
Conditional independence
*
Conditional event algebra A standard, Boolean algebra of events is a set of events related to one another by the familiar operations ''and'', ''or'', and ''not''. A conditional event algebra (CEA) contains not just ordinary events but also conditional events, which have the ...
**
Goodman–Nguyen–van Fraassen algebra
Random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s
Discrete and continuous random variables
*
Discrete random variables:
Probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
s
*
Continuous random variables:
Probability density functions
*
Normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.
...
s
*
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
s
*
Joint,
marginal and
conditional
Conditional (if then) may refer to:
* Causal conditional, if X then Y, where X is a cause of Y
* Conditional probability, the probability of an event A given that another event B has occurred
*Conditional proof, in logic: a proof that asserts a ...
distributions
Expectation
*
Expectation
Expectation or Expectations may refer to:
Science
* Expectation (epistemic)
* Expected value, in mathematical probability theory
* Expectation value (quantum mechanics)
* Expectation–maximization algorithm, in statistics
Music
* ''Expectation' ...
(or
mean),
variance and
covariance
**
Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
* General
moments about the mean
*
Correlated and
uncorrelated random variables
*
Conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
:
**
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
*
Independent random variables
Some common distributions
* Discrete:
**
constant (see also
degenerate distribution),
**
Bernoulli and
binomial,
**
negative binomial,
**
(discrete) uniform,
**
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
,
**
Poisson, and
**
hypergeometric.
* Continuous:
**
(continuous) uniform,
**
exponential,
**
gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
,
**
beta
Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiod ...
,
**
normal (or ''Gaussian'') and
multivariate normal,
**
χ-squared (or chi-squared),
**
F-distribution,
**
Student's t-distribution
In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in sit ...
, and
**
Cauchy.
Some other distributions
*
Cantor
*
Fisher–Tippett (or ''Gumbel'')
*
Pareto
*
Benford's law
Functions of random variables
*
Sum of normally distributed random variables
*
Borel's paradox
Generating functions
(Related topics:
integral transforms)
Common generating functions
*
Probability-generating function In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are oft ...
s
*
Moment-generating functions
*
Laplace transforms and
Laplace–Stieltjes transforms
*
Characteristic functions
Applications
*
A proof of the central limit theorem
Convergence of random variables
(Related topics:
convergence)
Modes of convergence
*
Convergence in distribution and
convergence in probability,
* Convergence in
mean,
mean square and
''r''th 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
**
Berry–Esséen theorem
*
Law of the iterated logarithm
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924). Another statement was given by A ...
Stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es
Some common
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es
*
Random walk
*
Poisson process
*
Compound Poisson process
*
Wiener process
*
Geometric Brownian motion
*
Fractional Brownian motion
*
Brownian bridge
A Brownian bridge is a continuous-time stochastic process ''B''(''t'') whose probability distribution is the conditional probability distribution of a standard Wiener process ''W''(''t'') (a mathematical model of Brownian motion) subject to the co ...
*
Ornstein–Uhlenbeck process
*
Gamma process
Markov processes
*
Markov property
*
Branching process
**
Galton–Watson process
*
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
**
Examples of Markov chains
*
Population processes
* Applications to
queueing theory
**
Erlang distribution
Stochastic differential equations
*
Stochastic calculus
*
Diffusions
**
Brownian motion
**
Wiener equation
A simple mathematical representation of Brownian motion, the Wiener equation, named after Norbert Wiener, assumes the current velocity of a fluid particle fluctuates randomly:
:\mathbf = \frac = g(t),
where v is velocity, x is position, ''d/dt'' ...
**
Wiener process
Time series
*
Moving-average and
autoregressive processes
*
Correlation function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
and
autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
Martingales
*
Martingale central limit theorem
*
Azuma's inequality
See also
*
Catalog of articles in probability theory
This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variab ...
*
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
{{DEFAULTSORT:Outline of probability
Probability
Probability
*
Probability
Probability