Related concepts and fundamentals:
Agnosticism
Epistemology
Presupposition
Probability
v
t
e
Probability
Probability is the measure of the likelihood that an event will
occur.[1] See glossary of probability and statistics.
Probability
Probability is
quantified as a number between 0 and 1, where, loosely speaking,[2] 0
indicates impossibility and 1 indicates certainty.[3][4] The higher
the probability of an event, the more likely it is that the event will
occur. A simple example is the tossing of a fair (unbiased) coin.
Since the coin is fair, the two outcomes ("heads" and "tails") are
both equally probable; the probability of "heads" equals the
probability of "tails"; and since no other outcomes are possible, the
probability of either "heads" or "tails" is 1/2 (which could also be
written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization
in probability theory, which is used widely in such areas of study as
mathematics, statistics, finance, gambling, science (in particular
physics), artificial intelligence/machine learning, computer science,
game theory, and philosophy to, for example, draw inferences about the
expected frequency of events.
Probability theory
Probability theory is also used to
describe the underlying mechanics and regularities of complex
systems.[5]
Contents
1 Interpretations
2 Etymology
3 History
4 Theory
5 Applications
6 Mathematical treatment
6.1 Independent events
6.2 Mutually exclusive events
6.3 Conditional probability
6.4 Inverse probability
6.5 Summary of probabilities
7 Relation to randomness and probability in quantum mechanics
8 See also
9 Notes
10 Bibliography
11 External links
Interpretations[edit]
Main article:
Probability
Probability interpretations
When dealing with experiments that are random and well-defined in a
purely theoretical setting (like tossing a fair coin), probabilities
can be numerically described by the number of desired outcomes divided
by the total number of all outcomes. For example, tossing a fair coin
twice will yield "head-head", "head-tail", "tail-head", and
"tail-tail" outcomes. The probability of getting an outcome of
"head-head" is 1 out of 4 outcomes or 1/4 or 0.25 (or 25%). When it
comes to practical application however, there are two major competing
categories of probability interpretations, whose adherents possess
different views about the fundamental nature of probability:
Objectivists assign numbers to describe some objective or physical
state of affairs. The most popular version of objective probability is
frequentist probability, which claims that the probability of a random
event denotes the relative frequency of occurrence of an experiment's
outcome, when repeating the experiment. This interpretation considers
probability to be the relative frequency "in the long run" of
outcomes.[6] A modification of this is propensity probability, which
interprets probability as the tendency of some experiment to yield a
certain outcome, even if it is performed only once.
Subjectivists assign numbers per subjective probability, i.e., as a
degree of belief.[7] The degree of belief has been interpreted as,
"the price at which you would buy or sell a bet that pays 1 unit of
utility if E, 0 if not E."[8] The most popular version of subjective
probability is Bayesian probability, which includes expert knowledge
as well as experimental data to produce probabilities. The expert
knowledge is represented by some (subjective) prior probability
distribution. These data are incorporated in a likelihood function.
The product of the prior and the likelihood, normalized, results in a
posterior probability distribution that incorporates all the
information known to date.[9] By Aumann's agreement theorem, Bayesian
agents whose prior beliefs are similar will end up with similar
posterior beliefs. However, sufficiently different priors can lead to
different conclusions regardless of how much information the agents
share.[10]
Etymology[edit]
See also:
History of probability
History of probability § Etymology
Further information: Likelihood
The word probability derives from the Latin probabilitas, which can
also mean "probity", a measure of the authority of a witness in a
legal case in Europe, and often correlated with the witness's
nobility. In a sense, this differs much from the modern meaning of
probability, which, in contrast, is a measure of the weight of
empirical evidence, and is arrived at from inductive reasoning and
statistical inference.[11]
History[edit]
Main article: History of probability
The scientific study of probability is a modern development of
mathematics.
Gambling
Gambling shows that there has been an interest in
quantifying the ideas of probability for millennia, but exact
mathematical descriptions arose much later. There are reasons for the
slow development of the mathematics of probability. Whereas games of
chance provided the impetus for the mathematical study of probability,
fundamental issues[clarification needed] are still obscured by the
superstitions of gamblers.[12]
Christiaan Huygens
Christiaan Huygens likely published the first book on probability
According to Richard Jeffrey, "Before the middle of the seventeenth
century, the term 'probable' (Latin probabilis) meant approvable, and
was applied in that sense, unequivocally, to opinion and to action. A
probable action or opinion was one such as sensible people would
undertake or hold, in the circumstances."[13] However, in legal
contexts especially, 'probable' could also apply to propositions for
which there was good evidence.[14]
Gerolamo Cardano
The sixteenth century Italian polymath
Gerolamo Cardano
Gerolamo Cardano demonstrated
the efficacy of defining odds as the ratio of favourable to
unfavourable outcomes (which implies that the probability of an event
is given by the ratio of favourable outcomes to the total number of
possible outcomes[15]). Aside from the elementary work by Cardano, the
doctrine of probabilities dates to the correspondence of Pierre de
Fermat and
Blaise Pascal
Blaise Pascal (1654).
Christiaan Huygens
Christiaan Huygens (1657) gave the
earliest known scientific treatment of the subject.[16] Jakob
Bernoulli's
Ars Conjectandi
Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's
Doctrine of Chances (1718) treated the subject as a branch of
mathematics.[17] See Ian Hacking's The Emergence of Probability[11]
and James Franklin's The
Science
Science of Conjecture[18] for histories of
the early development of the very concept of mathematical probability.
The theory of errors may be traced back to Roger Cotes's Opera
Miscellanea (posthumous, 1722), but a memoir prepared by Thomas
Simpson in 1755 (printed 1756) first applied the theory to the
discussion of errors of observation.[citation needed] The reprint
(1757) of this memoir lays down the axioms that positive and negative
errors are equally probable, and that certain assignable limits define
the range of all errors. Simpson also discusses continuous errors and
describes a probability curve.
The first two laws of error that were proposed both originated with
Pierre-Simon Laplace. The first law was published in 1774 and stated
that the frequency of an error could be expressed as an exponential
function of the numerical magnitude of the error, disregarding sign.
The second law of error was proposed in 1778 by
Laplace
Laplace and stated
that the frequency of the error is an exponential function of the
square of the error.[19] The second law of error is called the normal
distribution or the Gauss law. "It is difficult historically to
attribute that law to Gauss, who in spite of his well-known precocity
had probably not made this discovery before he was two years old."[19]
Daniel Bernoulli
Daniel Bernoulli (1778) introduced the principle of the maximum
product of the probabilities of a system of concurrent errors.
Carl Friedrich Gauss
Adrien-Marie Legendre
Adrien-Marie Legendre (1805) developed the method of least squares,
and introduced it in his Nouvelles méthodes pour la détermination
des orbites des comètes (New Methods for Determining the Orbits of
Comets).[20] In ignorance of Legendre's contribution, an
Irish-American writer, Robert Adrain, editor of "The Analyst" (1808),
first deduced the law of facility of error,
ϕ
(
x
)
=
c
e
−
h
2
x
2
,
displaystyle phi (x)=ce^ -h^ 2 x^ 2 ,
where
h
displaystyle h
is a constant depending on precision of observation, and
c
displaystyle c
is a scale factor ensuring that the area under the curve equals 1. He
gave two proofs, the second being essentially the same as John
Herschel's (1850).[citation needed] Gauss gave the first proof that
seems to have been known in
Europe
Europe (the third after Adrain's) in 1809.
Further proofs were given by
Laplace
Laplace (1810, 1812), Gauss (1823), James
Ivory (1825, 1826), Hagen (1837),
Friedrich Bessel
Friedrich Bessel (1838), W. F.
Donkin (1844, 1856), and
Morgan Crofton (1870). Other contributors
were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni
Schiaparelli (1875). Peters's (1856) formula[clarification needed] for
r, the probable error of a single observation, is well known.[to
whom?]
In the nineteenth century authors on the general theory included
Laplace,
Sylvestre Lacroix
Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet
(1853),
Richard Dedekind
Richard Dedekind (1860), Helmert (1872), Hermann Laurent
(1873), Liagre, Didion, and Karl Pearson.
Augustus De Morgan
Augustus De Morgan and
George Boole
George Boole improved the exposition of the theory.
Andrey Markov
Andrey Markov introduced[21] the notion of
Markov chains
Markov chains (1906), which
played an important role in stochastic processes theory and its
applications. The modern theory of probability based on the measure
theory was developed by
Andrey Kolmogorov
Andrey Kolmogorov (1931).[22]
On the geometric side (see integral geometry) contributors to The
Educational Times were influential (Miller, Crofton, McColl,
Wolstenholme, Watson, and Artemas Martin).[citation needed]
Further information: History of statistics
Theory[edit]
Main article:
Probability
Probability theory
Like other theories, the theory of probability is a representation of
its concepts in formal terms—that is, in terms that can be
considered separately from their meaning. These formal terms are
manipulated by the rules of mathematics and logic, and any results are
interpreted or translated back into the problem domain.
There have been at least two successful attempts to formalize
probability, namely the
Kolmogorov
Kolmogorov formulation and the Cox
formulation. In Kolmogorov's formulation (see probability space), sets
are interpreted as events and probability itself as a measure on a
class of sets. In Cox's theorem, probability is taken as a primitive
(that is, not further analyzed) and the emphasis is on constructing a
consistent assignment of probability values to propositions. In both
cases, the laws of probability are the same, except for technical
details.
There are other methods for quantifying uncertainty, such as the
Dempster–Shafer theory
Dempster–Shafer theory or possibility theory, but those are
essentially different and not compatible with the laws of probability
as usually understood.
Applications[edit]
Probability theory
Probability theory is applied in everyday life in risk assessment and
modeling. The insurance industry and markets use actuarial science to
determine pricing and make trading decisions. Governments apply
probabilistic methods in environmental regulation, entitlement
analysis (Reliability theory of aging and longevity), and financial
regulation.
A good example of the use of probability theory in equity trading is
the effect of the perceived probability of any widespread Middle East
conflict on oil prices, which have ripple effects in the economy as a
whole. An assessment by a commodity trader that a war is more likely
can send that commodity's prices up or down, and signals other traders
of that opinion. Accordingly, the probabilities are neither assessed
independently nor necessarily very rationally. The theory of
behavioral finance emerged to describe the effect of such groupthink
on pricing, on policy, and on peace and conflict.[23]
In addition to financial assessment, probability can be used to
analyze trends in biology (e.g. disease spread) as well as ecology
(e.g. biological Punnett squares). As with finance, risk assessment
can be used as a statistical tool to calculate the likelihood of
undesirable events occurring and can assist with implementing
protocols to avoid encountering such circumstances.
Probability
Probability is
used to design games of chance so that casinos can make a guaranteed
profit, yet provide payouts to players that are frequent enough to
encourage continued play.[24]
The discovery of rigorous methods to assess and combine probability
assessments has changed society.[25][citation needed] It is important
for most citizens to understand how probability assessments are made,
and how they contribute to decisions.[25][citation needed]
Another significant application of probability theory in everyday life
is reliability. Many consumer products, such as automobiles and
consumer electronics, use reliability theory in product design to
reduce the probability of failure. Failure probability may influence a
manufacturer's decisions on a product's warranty.[26]
The cache language model and other statistical language models that
are used in natural language processing are also examples of
applications of probability theory.
Mathematical treatment[edit]
See also:
Probability
Probability axioms
Consider an experiment that can produce a number of results. The
collection of all possible results is called the sample space of the
experiment. The power set of the sample space is formed by considering
all different collections of possible results. For example, rolling a
dice can produce six possible results. One collection of possible
results gives an odd number on the dice. Thus, the subset 1,3,5 is
an element of the power set of the sample space of dice rolls. These
collections are called "events". In this case, 1,3,5 is the event
that the dice falls on some odd number. If the results that actually
occur fall in a given event, the event is said to have occurred.
A 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 1,2,3,4,5,6 ) is assigned a value
of one. To qualify as a probability, the assignment of values must
satisfy the requirement that if you look at a collection of mutually
exclusive events (events with no common results, e.g., the events 1,6
, 3 , and 2,4 are all mutually exclusive), the probability that at
least one of the events will occur is given by the sum of the
probabilities of all the individual events.[27]
The probability of an event A is written as
P
(
A
)
displaystyle P(A)
,
p
(
A
)
displaystyle p(A)
, or
Pr
(
A
)
displaystyle text Pr (A)
.[28] This mathematical definition of probability can extend to
infinite sample spaces, and even uncountable sample spaces, using the
concept of a measure.
The opposite or complement of an event A is the event [not A] (that
is, the event of A not occurring), often denoted as
A
¯
,
A
∁
,
¬
A
displaystyle overline A ,A^ complement ,neg A
, or
∼
A
displaystyle sim A
; its probability is given by P(not A) = 1 − P(A).[29] As an
example, the chance of not rolling a six on a six-sided die is 1 –
(chance of rolling a six)
=
1
−
1
6
=
5
6
displaystyle =1- tfrac 1 6 = tfrac 5 6
. See
Complementary event
Complementary event for a more complete treatment.
If two events A and B occur on a single performance of an experiment,
this is called the intersection or joint probability of A and B,
denoted as
P
(
A
∩
B
)
displaystyle P(Acap B)
.
Independent events[edit]
If two events, A and B are independent then the joint probability is
P
(
A
and
B
)
=
P
(
A
∩
B
)
=
P
(
A
)
P
(
B
)
,
displaystyle P(A mbox and B)=P(Acap B)=P(A)P(B),,
for example, if two coins are flipped the chance of both being heads
is
1
2
×
1
2
=
1
4
displaystyle tfrac 1 2 times tfrac 1 2 = tfrac 1 4
.[30]
Mutually exclusive events[edit]
If either event A or event B but never both occurs on a single
performance of an experiment, then they are called mutually exclusive
events.
If two events are mutually exclusive then the probability of both
occurring is denoted as
P
(
A
∩
B
)
displaystyle P(Acap B)
.
P
(
A
and
B
)
=
P
(
A
∩
B
)
=
0
displaystyle P(A mbox and B)=P(Acap B)=0
If two events are mutually exclusive then the probability of either
occurring is denoted as
P
(
A
∪
B
)
displaystyle P(Acup B)
.
P
(
A
or
B
)
=
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
=
P
(
A
)
+
P
(
B
)
−
0
=
P
(
A
)
+
P
(
B
)
displaystyle P(A mbox or B)=P(Acup B)=P(A)+P(B)-P(Acap
B)=P(A)+P(B)-0=P(A)+P(B)
For example, the chance of rolling a 1 or 2 on a six-sided die is
P
(
1
or
2
)
=
P
(
1
)
+
P
(
2
)
=
1
6
+
1
6
=
1
3
.
displaystyle P(1 mbox or 2)=P(1)+P(2)= tfrac 1 6 + tfrac 1
6 = tfrac 1 3 .
Conditional probability[edit]
Conditional probability
Conditional probability is the probability of some event A, given the
occurrence of some other event B.
Conditional probability
Conditional probability is written
P
(
A
∣
B
)
displaystyle P(Amid B)
, and is read "the probability of A, given B". It is defined by[31]
P
(
A
∣
B
)
=
P
(
A
∩
B
)
P
(
B
)
.
displaystyle P(Amid B)= frac P(Acap B) P(B) .,
If
P
(
B
)
=
0
displaystyle P(B)=0
then
P
(
A
∣
B
)
displaystyle P(Amid B)
is formally undefined by this expression. However, it is possible to
define a conditional probability for some zero-probability events
using a σ-algebra of such events (such as those arising from a
continuous random variable).[citation needed]
For example, in a bag of 2 red balls and 2 blue balls (4 balls in
total), the probability of taking a red ball is
1
/
2
displaystyle 1/2
; however, when taking a second ball, the probability of it being
either a red ball or a blue ball depends on the ball previously taken,
such as, if a red ball was taken, the probability of picking a red
ball again would be
1
/
3
displaystyle 1/3
since only 1 red and 2 blue balls would have been remaining.
Inverse probability[edit]
In probability theory and applications,
Bayes' rule
Bayes' rule relates the odds
of event
A
1
displaystyle A_ 1
to event
A
2
displaystyle A_ 2
, before (prior to) and after (posterior to) conditioning on another
event
B
displaystyle B
. The odds on
A
1
displaystyle A_ 1
to event
A
2
displaystyle A_ 2
is simply the ratio of the probabilities of the two events. When
arbitrarily many events
A
displaystyle A
are of interest, not just two, the rule can be rephrased as posterior
is proportional to prior times likelihood,
P
(
A
B
)
∝
P
(
A
)
P
(
B
A
)
displaystyle P(AB)propto P(A)P(BA)
where the proportionality symbol means that the left hand side is
proportional to (i.e., equals a constant times) the right hand side as
A
displaystyle A
varies, for fixed or given
B
displaystyle B
(Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to
Laplace
Laplace (1774) and to Cournot (1843); see Fienberg (2005). See Inverse
probability and Bayes' rule.
Summary of probabilities[edit]
Summary of probabilities
Event
Probability
A
P
(
A
)
∈
[
0
,
1
]
displaystyle P(A)in [0,1],
not A
P
(
A
∁
)
=
1
−
P
(
A
)
displaystyle P(A^ complement )=1-P(A),
A or B
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
if A and B are mutually exclusive
displaystyle begin aligned P(Acup B)&=P(A)+P(B)-P(Acap
B)\P(Acup B)&=P(A)+P(B)qquad mbox if A and B are mutually
exclusive \end aligned
A and B
P
(
A
∩
B
)
=
P
(
A
B
)
P
(
B
)
=
P
(
B
A
)
P
(
A
)
P
(
A
∩
B
)
=
P
(
A
)
P
(
B
)
if A and B are independent
displaystyle begin aligned P(Acap
B)&=P(AB)P(B)=P(BA)P(A)\P(Acap B)&=P(A)P(B)qquad mbox if A
and B are independent \end aligned
A given B
P
(
A
∣
B
)
=
P
(
A
∩
B
)
P
(
B
)
=
P
(
B
A
)
P
(
A
)
P
(
B
)
displaystyle P(Amid B)= frac P(Acap B) P(B) = frac P(BA)P(A)
P(B) ,
Relation to randomness and probability in quantum mechanics[edit]
Main article: Randomness
See also: Quantum fluctuation § Interpretations
This section needs expansion. You can help by adding to it. (April
2017)
In a deterministic universe, based on Newtonian concepts, there would
be no probability if all conditions were known (Laplace's demon), (but
there are situations in which sensitivity to initial conditions
exceeds our ability to measure them, i.e. know them). In the case of a
roulette wheel, if the force of the hand and the period of that force
are known, the number on which the ball will stop would be a certainty
(though as a practical matter, this would likely be true only of a
roulette wheel that had not been exactly levelled – as Thomas A.
Bass' Newtonian Casino revealed). This also assumes knowledge of
inertia and friction of the wheel, weight, smoothness and roundness of
the ball, variations in hand speed during the turning and so forth. A
probabilistic description can thus be more useful than Newtonian
mechanics for analyzing the pattern of outcomes of repeated rolls of a
roulette wheel. Physicists face the same situation in kinetic theory
of gases, where the system, while deterministic in principle, is so
complex (with the number of molecules typically the order of magnitude
of the
Avogadro constant
Avogadro constant 7023602000000000000♠6.02×1023) that only a
statistical description of its properties is feasible.
Probability theory
Probability theory is required to describe quantum phenomena.[32] A
revolutionary discovery of early 20th century physics was the random
character of all physical processes that occur at sub-atomic scales
and are governed by the laws of quantum mechanics. The objective wave
function evolves deterministically but, according to the Copenhagen
interpretation, it deals with probabilities of observing, the outcome
being explained by a wave function collapse when an observation is
made. However, the loss of determinism for the sake of instrumentalism
did not meet with universal approval.
Albert Einstein
Albert Einstein famously
remarked in a letter to Max Born: "I am convinced that God does not
play dice".[33] Like Einstein, Erwin Schrödinger, who discovered the
wave function, believed quantum mechanics is a statistical
approximation of an underlying deterministic reality.[34] In some
modern interpretations of the statistical mechanics of measurement,
quantum decoherence is invoked to account for the appearance of
subjectively probabilistic experimental outcomes.
See also[edit]
Mathematics
Mathematics portal
Logic
Logic portal
Main article: Outline of probability
Chance (other)
Class membership probabilities
Equiprobability
Heuristics in judgment and decision-making
Probability
Probability theory
Statistics
Estimators
Estimation Theory
Probability
Probability density function
In Law
Balance of probabilities
Notes[edit]
^ "Probability". Webster's Revised Unabridged Dictionary. G & C
Merriam, 1913
^ Strictly speaking, a probability of 0 indicates that an event almost
never takes place, whereas a probability of 1 indicates than an event
almost certainly takes place. This is an important distinction when
the sample space is infinite. For example, for the continuous uniform
distribution on the real interval [5, 10], there are an infinite
number of possible outcomes, and the probability of any given outcome
being observed — for instance, exactly 7 — is 0. This means that
when we make an observation, it will almost surely not be exactly 7.
However, it does not mean that exactly 7 is impossible. Ultimately
some specific outcome (with probability 0) will be observed, and one
possibility for that specific outcome is exactly 7.
^ "Kendall's Advanced
Theory
Theory of Statistics, Volume 1: Distribution
Theory", Alan Stuart and Keith Ord, 6th Ed, (2009),
ISBN 9780534243128
^ William Feller, "An Introduction to
Probability
Probability
Theory
Theory and Its
Applications", (Vol 1), 3rd Ed, (1968), Wiley, ISBN 0-471-25708-7
^
Probability
Probability
Theory
Theory The Britannica website
^ Hacking, Ian (1965). The
Logic
Logic of
Statistical
Statistical Inference. Cambridge
University Press. ISBN 0-521-05165-7. [page needed]
^ Finetti, Bruno de (1970). "Logical foundations and measurement of
subjective probability". Acta Psychologica. 34: 129–145.
doi:10.1016/0001-6918(70)90012-0.
^ Hájek, Alan. "Interpretations of Probability". The Stanford
Encyclopedia of
Philosophy
Philosophy (Winter 2012 Edition), Edward N. Zalta
(ed.). Retrieved 22 April 2013.
^ Hogg, Robert V.; Craig, Allen; McKean, Joseph W. (2004).
Introduction to Mathematical
Statistics
Statistics (6th ed.). Upper Saddle River:
Pearson. ISBN 0-13-008507-3. [page needed]
^ Jaynes, E. T. (2003-06-09). "Section 5.3 Converging and diverging
views". In Bretthorst, G. Larry.
Probability
Probability Theory: The
Logic
Logic of
Science
Science (1 ed.). Cambridge University Press.
ISBN 9780521592710.
^ a b Hacking, I. (2006) The Emergence of Probability: A Philosophical
Study of Early Ideas about Probability, Induction and Statistical
Inference, Cambridge University Press,
ISBN 978-0-521-68557-3[page needed]
^ Freund, John. (1973) Introduction to Probability. Dickenson
ISBN 978-0822100782 (p. 1)
^ Jeffrey, R.C.,
Probability
Probability and the Art of Judgment, Cambridge
University Press. (1992). pp. 54–55 . ISBN 0-521-39459-7
^ Franklin, J. (2001) The
Science
Science of Conjecture: Evidence and
Probability
Probability Before Pascal, Johns Hopkins University Press. (pp. 22,
113, 127)
^ Some laws and problems in classical probability and how Cardano
anticipated them Gorrochum, P. Chance magazine 2012
^ Abrams, William, A Brief History of Probability, Second Moment,
retrieved 2008-05-23
^ Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008). Quantum
leap : from Dirac and Feynman, across the universe, to human body
and mind. Singapore ; Hackensack, NJ: World Scientific.
p. 16. ISBN 978-981-281-927-7.
^ Franklin, James (2001). The
Science
Science of Conjecture: Evidence and
Probability
Probability Before Pascal. Johns Hopkins University Press.
ISBN 0801865697.
^ a b Wilson EB (1923) "First and second laws of error". Journal of
the American
Statistical
Statistical Association, 18, 143
^ Seneta, Eugene William. ""Adrien-Marie Legendre" (version 9)".
StatProb: The Encyclopedia Sponsored by
Statistics
Statistics and Probability
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Bibliography[edit]
Kallenberg, O. (2005) Probabilistic Symmetries and Invariance
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Kallenberg, O. (2002) Foundations of Modern Probability, 2nd ed.
Springer Series in Statistics. 650 pp. ISBN 0-387-95313-2
Olofsson, Peter (2005) Probability, Statistics, and Stochastic
Processes, Wiley-Interscience. 504 pp ISBN 0-471-67969-0.
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