Probability theory or probability calculus is the branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
concerned with
probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
. Although there are several different
probability interpretations
The word "probability" has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly on ...
, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of
axioms. Typically these axioms formalise probability in terms of a
probability space, which assigns a
measure taking values between 0 and 1, termed the
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
, to a set of outcomes called the
sample space. Any specified subset of the sample space is called an
event.
Central subjects in probability theory include discrete and continuous
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s,
probability distributions, and
stochastic processes (which provide mathematical abstractions of
non-deterministic or uncertain processes or measured
quantities that may either be single occurrences or evolve over time in a random fashion).
Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the
law of large numbers and the
central limit theorem.
As a mathematical foundation for
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
or
sequential estimation. A great discovery of twentieth-century
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
was the probabilistic nature of physical phenomena at atomic scales, described in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.
History of probability
The modern mathematical theory of
probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
has its roots in attempts to analyze
games of chance by
Gerolamo Cardano in the sixteenth century, and by
Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
and
Blaise Pascal in the seventeenth century (for example the "
problem of points").
Christiaan Huygens
Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
published a book on the subject in 1657. In the 19th century, what is considered the
classical definition of probability was completed by
Pierre Laplace.
Initially, probability theory mainly considered events, and its methods were mainly
combinatorial. Eventually,
analytical considerations compelled the incorporation of variables into the theory.
This culminated in modern probability theory, on foundations laid by
Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of
sample space, introduced by
Richard von Mises, and
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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
and presented his
axiom system for probability theory in 1933. This became the mostly undisputed
axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by
Bruno de Finetti
Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 , which discuss ...
.
Treatment
Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more.
Motivation
Consider an
experiment that can produce a number of outcomes. The set of all outcomes is called the ''
sample space'' of the experiment. The ''
power set'' of the sample space (or equivalently, the event space) is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of dice rolls. These collections are called ''events''. In this case, is the event that the die falls on some odd number. If the results that actually occur fall in a given event, that event is said to have occurred.
Probability is a
way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event ) be assigned a value of one. To qualify as a
probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events , , and are all mutually exclusive), the probability that any of these events occurs is given by the sum of the probabilities of the events.
The probability that any one of the events , , or will occur is 5/6. This is the same as saying that the probability of event is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, that is, absolute certainty.
When doing calculations using the outcomes of an experiment, it is necessary that all those
elementary events have a number assigned to them. This is done using a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
. A random variable is a function that assigns to each elementary event in the sample space a
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
. This function is usually denoted by a capital letter. In the case of a die, the assignment of a number to certain elementary events can be done using the
identity function. This does not always work. For example, when
flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable ''X'' could assign to the outcome "heads" the number "0" (
) and to the outcome "tails" the number "1" (
).
Discrete probability distributions
deals with events that occur in
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
sample spaces.
Examples: Throwing
dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, ro ...
, experiments with
decks of cards,
random walk, and tossing
coin
A coin is a small object, usually round and flat, used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to facilitate trade. They are most often issued by ...
s.
:
Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see
Classical definition of probability.
For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by
, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.
:
The modern definition starts with a
finite or countable set called the
sample space, which relates to the set of all ''possible outcomes'' in classical sense, denoted by
. It is then assumed that for each element
, an intrinsic "probability" value
is attached, which satisfies the following properties:
#
#
That is, the probability function ''f''(''x'') lies between zero and one for every value of ''x'' in the sample space ''Ω'', and the sum of ''f''(''x'') over all values ''x'' in the sample space ''Ω'' is equal to 1. An is defined as any
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of the sample space
. The of the event
is defined as
:
So, the probability of the entire sample space is 1, and the probability of the null event is 0.
The function
mapping a point in the sample space to the "probability" value is called a abbreviated as .
Continuous probability distributions

deals with events that occur in a continuous sample space.
:
The classical definition breaks down when confronted with the continuous case. See
Bertrand's paradox.
:
If the sample space of a random variable ''X'' is the set of
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
(
) or a subset thereof, then a function called the ()
exists, defined by
. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''.
The CDF necessarily satisfies the following properties.
#
is a
monotonically non-decreasing,
right-continuous function;
#
#
The random variable
is said to have a continuous probability distribution if the corresponding CDF
is continuous. If
is
absolutely continuous, then its derivative exists almost everywhere and integrating the derivative gives us the CDF back again. In this case, the random variable ''X'' is said to have a () or simply
For a set
, the probability of the random variable ''X'' being in
is
:
In case the PDF exists, this can be written as
:
Whereas the ''PDF'' exists only for continuous random variables, the ''CDF'' exists for all random variables (including discrete random variables) that take values in
These concepts can be generalized for
multidimensional cases on
and other continuous sample spaces.
Measure-theoretic probability theory
The utility of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.
An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a PDF of
, where
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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
to define the
probability space:
Given any set
(also called ) and a
σ-algebra on it, a
measure defined on
is called a if
If
is the
Borel σ-algebra on the set of real numbers, then there is a unique probability measure on
for any CDF, and vice versa. The measure corresponding to a CDF is said to be by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies.
The ''probability'' of a set
in the σ-algebra
is defined as
:
where the integration is with respect to the measure
induced by
Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside
, as in the theory of
stochastic processes. For example, to study
Brownian motion, probability is defined on a space of functions.
When it is convenient to work with a dominating measure, the
Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure. Discrete densities are usually defined as this derivative with respect to a
counting measure over the set of all possible outcomes. Densities for
absolutely continuous distributions are usually defined as this derivative with respect to the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
. If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions.
Classical probability distributions
Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions, therefore, have gained ''special importance'' in probability theory. Some fundamental ''discrete distributions'' are the
discrete uniform,
Bernoulli,
binomial,
negative binomial,
Poisson and
geometric distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
* The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \;
* T ...
s. Important ''continuous distributions'' include the
continuous uniform,
normal,
exponential,
gamma
Gamma (; uppercase , lowercase ; ) 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 normally repr ...
and
beta distributions.
Convergence of random variables
In probability theory, there are several notions of convergence for
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions.
;Weak convergence: A sequence of random variables
converges to the random variable
if their respective CDF converges
converges to the CDF
of
, wherever
is
continuous. Weak convergence is also called .
:Most common shorthand notation:
;Convergence in probability: The sequence of random variables
is said to converge towards the random variable
if
for every ε > 0.
:Most common shorthand notation:
;Strong convergence: The sequence of random variables
is said to converge towards the random variable
if
. Strong convergence is also known as .
:Most common shorthand notation:
As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies weak convergence. The reverse statements are not always true.
Law of large numbers
Common intuition suggests that if a fair coin is tossed many times, then ''roughly'' half of the time it will turn up ''heads'', and the other half it will turn up ''tails''. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of ''heads'' to the number of ''tails'' will approach unity. Modern probability theory provides a formal version of this intuitive idea, known as the . This law is remarkable because it is not assumed in the foundations of probability theory, but instead emerges from these foundations as a theorem. Since it links theoretically derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory and has had widespread influence.
The (LLN) states that the sample average
:
of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
independent and identically distributed random variables converges towards their common
expectation (expected value)
, provided that the expectation of
is finite.
It is in the different forms of
convergence of random variables that separates the ''weak'' and the ''strong'' law of large numbers
:Weak law:
for
:Strong law:
for
It follows from the LLN that if an event of probability ''p'' is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards ''p''.
For example, if
are independent
Bernoulli random variables taking values 1 with probability ''p'' and 0 with probability 1-''p'', then
for all ''i'', so that
converges to ''p''
almost surely.
Central limit theorem
The central limit theorem (CLT) explains the ubiquitous occurrence of the
normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
f(x) = \frac ...
in nature, and this theorem, according to David Williams, "is one of the great results of mathematics."
[ David Williams, "Probability with martingales", Cambridge 1991/2008]
The theorem states that the
average
In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
of many independent and identically distributed random variables with finite variance tends towards a normal distribution ''irrespective'' of the distribution followed by the original random variables. Formally, let
be independent random variables with
mean and
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
Then the sequence of random variables
:
converges in distribution to a
standard normal random variable.
For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the
Berry–Esseen theorem. For example, the distributions with finite first, second, and third moment from the
exponential family; on the other hand, for some random variables of the
heavy tail and
fat tail variety, it works very slowly or may not work at all: in such cases one may use the
Generalized Central Limit Theorem (GCLT).
See also
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Lists
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References
Citations
Sources
*
:: The first major treatise blending calculus with probability theory, originally in French: ''Théorie Analytique des Probabilités''.
*
:: An English translation by Nathan Morrison appeared under the title ''Foundations of the Theory of Probability'' (Chelsea, New York) in 1950, with a second edition in 1956.
*
*
Olav Kallenberg; ''Foundations of Modern Probability,'' 2nd ed. Springer Series in Statistics. (2002). 650 pp.
*
:: A lively introduction to probability theory for the beginner.
* Olav Kallenberg; ''Probabilistic Symmetries and Invariance Principles''. Springer -Verlag, New York (2005). 510 pp.
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{{DEFAULTSORT:Probability Theory