Probability theory: Difference between revisions

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{{Probability fundamentals}}

'''Probability theory''' or '''probability calculus''' is the branch of [[mathematics]] concerned with [[probability]]. Although there are several different [[probability interpretations]], probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of [[axioms of probability|axioms]]. Typically these axioms formalise probability in terms of a [[probability space]], which assigns a [[measure (mathematics)|measure]] taking values between 0 and 1, termed the [[probability measure]], to a set of outcomes called the [[sample space]]. Any specified subset of the sample space is called an [[event (probability theory)|event]].

Central subjects in probability theory include discrete and continuous [[random variable]]s, [[probability distributions]], and [[stochastic process]]es (which provide mathematical abstractions of [[determinism|non-deterministic]] or uncertain processes or measured [[Quantity|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]], probability theory is essential to many human activities that involve quantitative analysis of data.<ref>[http://home.ubalt.edu/ntsbarsh/stat-data/Topics.htm Inferring From Data]</ref> Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in [[statistical mechanics]] or [[sequential estimation]]. A great discovery of twentieth-century [[physics]] was the probabilistic nature of physical phenomena at atomic scales, described in [[quantum mechanics]]. <ref>{{cite encyclopedia |title=Quantum Logic and Probability Theory |encyclopedia=The Stanford Encyclopedia of Philosophy |date=10 August 2021|url= https://plato.stanford.edu/entries/qt-quantlog/ }}</ref>

==History of probability==

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===Motivation===

Consider an [[Experiment (probability theory)|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 {1,3,5} is an element of the power set of the sample space of diedice rolls. These collections are called ''events''. In this case, {1,3,5} 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 [[Function (mathematics)|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}) be assigned a value of one. To qualify as a [[probability distribution]], 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 {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that any of these events occurs is given by the sum of the probabilities of the events.<ref>{{cite book |last=Ross |first=Sheldon |title=A First Course in Probability |publisher=Pearson Prentice Hall |edition=8th |year=2010 |isbn=978-0-13-603313-4 |pages=26–27 |url=https://books.google.com/books?id=Bc1FAQAAIAAJ&pg=PA26 |access-date=2016-02-28 }}</ref>

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The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} 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 event]]s have a number assigned to them. This is done using a [[random variable]]. A random variable is a function that assigns to each elementary event in the sample space a [[real number]]. This function is usually denoted by a capital letter.<ref>{{Cite book |title =Introduction to Probability and Mathematical Statistics |last1 =Bain |first1 =Lee J. |last2 =Engelhardt |first2 =Max |publisher =Brooks/Cole |___location =[[Belmont, California]] |page =53 |isbn =978-0-534-38020-5 |edition =2nd |date =1992 }}</ref> In the case of a die, the assignment of a number to a certain elementary events can be done using the [[identity function]]. This does not always work. For example, when [[coin flipping|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" (<math display="inline">X(\text{heads})=0</math>) and to the outcome "tails" the number "1" (<math>X(\text{tails})=1</math>).

===Discrete probability distributions===

{{Main|Discrete probability distribution}}

[[File:NYW-DK-Poisson(5).svg|thumb|300px|The [[Poisson distribution]], a discrete probability distribution.]]

{{em|Discrete probability theory}} deals with events that occur in [[countable]] sample spaces.

Examples: Throwing [[dice]], experiments with [[deck of cards|decks of cards]], [[random walk]], and tossing [[coin]]s.

{{em|Classical definition}}:

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 diedice is rolled", the probability is given by <math>\tfrac{3}{6}=\tfrac{1}{2}</math>, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.

{{em|Modern definition}}:

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So, the probability of the entire sample space is 1, and the probability of the null event is 0.

The function <math>f(x)\,</math> mapping a point in the sample space to the "probability" value is called a {{em|probability mass function}} abbreviated as {{em|pmf}}. The modern definition does not try to answer how probability mass functions are obtained; instead, it builds a theory that assumes their existence{{citation needed|date=December 2015}}.

===Continuous probability distributions===

{{Main|Continuous probability distribution}}

[[File:Gaussian distribution 2.jpg|thumb|300px|The [[normal distribution]], a continuous probability distribution.]]

{{em|Continuous probability theory}} deals with events that occur in a continuous sample space.

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{{em|Modern definition}}:

If the sample space of a random variable ''X'' is the set of [[real numbers]] (<math>\mathbb{R}</math>) or a subset thereof, then a function called the {{em|[[cumulative distribution function]]}} (or {{em|cdfCDF}}) <math>F\,</math> exists, defined by <math>F(x) = P(X\le x) \,</math>. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''.

The cdfCDF necessarily satisfies the following properties.

# <math>F\,</math> is a [[Monotonic function|monotonically non-decreasing]], [[right-continuous]] function;

# <math>\lim_{x\rightarrow -\infty} F(x)=0\,;</math>

# <math>\lim_{x\rightarrow \infty} F(x)=1\,.</math>

The random variable <math>X</math> is said to have a continuous probability distribution if the corresponding cdfCDF <math>F</math> is continuous. If <math>F\,</math> is [[absolutely continuous]], i.e.,then its derivative exists almost everywhere and integrating the derivative gives us the cdfCDF back again,. thenIn this case, the random variable ''X'' is said to have a {{em|[[probability density function]]}} or ({{em|pdfPDF}}) or simply {{em|density}} <math>f(x)=\frac{dF(x)}{dx}\,.</math>

For a set <math>E \subseteq \mathbb{R}</math>, the probability of the random variable ''X'' being in <math>E\,</math> is

:<math>P(X\in E) = \int_{x\in E} dF(x)\,.</math>

In case the probability density functionPDF exists, this can be written as

:<math>P(X\in E) = \int_{x\in E} f(x)\,dx\,.</math>

Whereas the ''pdfPDF'' exists only for continuous random variables, the ''cdfCDF'' exists for all random variables (including discrete random variables) that take values in <math>\mathbb{R}\,.</math>

These concepts can be generalized for [[Dimension|multidimensional]] cases on <math>\mathbb{R}^n</math> and other continuous sample spaces.

===Measure-theoretic probability theory===

The ''[[wikt:raison d'être|raison d'être]]''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 pdfPDF of <math>(\delta[x] + \varphi(x))/2</math>, where <math>\delta[x]</math> is the [[Dirac delta function]].

Other distributions may not even be a mix, for example, the [[Cantor distribution]] has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using [[measure theory]] to define the [[probability space]]:

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Given any set <math>\Omega\,</math> (also called {{em|sample space}}) and a [[sigma-algebra|σ-algebra]] <math>\mathcal{F}\,</math> on it, a [[measure (mathematics)|measure]] <math>P\,</math> defined on <math>\mathcal{F}\,</math> is called a {{em|probability measure}} if <math>P(\Omega)=1.\,</math>

If <math>\mathcal{F}\,</math> is the [[Borel algebra|Borel σ-algebra]] on the set of real numbers, then there is a unique probability measure on <math>\mathcal{F}\,</math> for any cdfCDF, and vice versa. The measure corresponding to a cdfCDF is said to be {{em|induced}} by the cdfCDF. This measure coincides with the pmf for discrete variables and pdfPDF for continuous variables, making the measure-theoretic approach free of fallacies.

The ''probability'' of a set <math>E\,</math> in the σ-algebra <math>\mathcal{F}\,</math> is defined as

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Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside <math>\mathbb{R}^n</math>, as in the theory of [[stochastic process]]es. For example, to study [[Brownian motion]], probability is defined on a space of functions.

When it's is convenient to work with a dominating measure, the [[Radon-NikodymRadon–Nikodym theorem]] is used to define a density as the Radon-NikodymRadon–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]]. 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==

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In probability theory, there are several notions of convergence for [[random variable]]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 <math>X_1,X_2,\dots,\,</math> converges {{em|weakly}} to the random variable <math>X\,</math> if their respective cumulative ''distribution functions''CDF converges<math>F_1,F_2,\dots\,</math> convergeconverges to the cumulative distribution functionCDF <math>F\,</math> of <math>X\,</math>, wherever <math>F\,</math> is [[continuous function|continuous]]. Weak convergence is also called {{em|convergence in distribution}}.

:Most common shorthand notation: <math>\displaystyle X_n \, \xrightarrow{\mathcal D} \, X</math>

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The {{em|law of large numbers}} (LLN) states that the sample average

:<math>\overline{X}_n=\frac1n{\sum_{k=1}^n X_k}</math>

of a [[sequence]] of [[independent and identically distributed random variables]] <math>X_k</math> converges towards their common [[Expected value|expectation]] (expected value) <math>\mu</math>, provided that the expectation of <math>|X_k|</math> is finite.

of a sequence of independent and

identically distributed random variables <math>X_k</math> converges towards their common expectation <math>\mu</math>, provided that the expectation of <math>|X_k|</math> is finite.

It is in the different forms of [[convergence of random variables]] that separates the ''weak'' and the ''strong'' law of large numbers<ref>{{Cite book|last=Dekking|first=Michel|url=http://archive.org/details/modernintroducti00fmde|title=A modern introduction to probability and statistics : understanding why and how|date=2005|publisher=London : Springer|others=Library Genesis|isbn=978-1-85233-896-1|pages=180–194|chapter=Chapter 13: The law of large numbers}}</ref>

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==See also==

{{Portal|Mathematics}}

* {{Annotated link|Mathematical Statistics}}

{{cols|colwidth=21em}}

* {{Annotated link|Expected value}}

* [[Catalog of articles in probability theory]]

* [[Expected{{Annotated value]] and [[link|Variance]]}}

* [[{{Annotated link|Fuzzy logic]] and [[Fuzzy measure theory]]}}

* {{Annotated link|Fuzzy measure theory}}

* [[{{Annotated link|Glossary of probability and statistics]]}}

* [[{{Annotated link|Likelihood function]]}}

* [[List of probability topics]]

* [[{{Annotated link|Notation in probability]]}}

* [[List of publications in statistics]]

* [[{{Annotated link|Predictive modelling]]}}

* [[List of statistical topics]]

* [[{{Annotated link|Probabilistic logic]] – |fallback=A combination of probability theory and logic}}

* [[Notation in probability]]

* [[{{Annotated link|Probabilistic proofs of non-probabilistic theorems]]}}

* [[Predictive modelling]]

* {{Annotated link|Probability distribution}}

* [[Probabilistic logic]] – A combination of probability theory and logic

* {{Annotated link|Probability axioms}}

* [[Probabilistic proofs of non-probabilistic theorems]]

* [[{{Annotated link|Probability distribution]]interpretations}}

* [[{{Annotated link|Probability axioms]]space}}

* [[{{Annotated link|Statistical independence]]}}

* [[Probability interpretations]]

* [[{{Annotated link|Statistical physics]]}}

* [[Probability space]]

* [[{{Annotated link|Subjective logic]]}}

* [[Statistical independence]]

* [[{{Annotated link|Pairwise independence#Probability of the union of pairwise independent events|Pairwise independence§Probability of the union of pairwise independent events]]}}

* [[Statistical physics]]

=== Lists ===

* [[Subjective logic]]

* [[{{Annotated link|Catalog of articles in probability theory]]}}

* [[Pairwise independence#Probability of the union of pairwise independent events|Probability of the union of pairwise independent events]]

* [[{{Annotated link|List of probability topics]]}}

{{colend}}

* [[{{Annotated link|List of publications in statistics]]}}

* [[{{Annotated link|List of statistical topics]]}}

== References ==

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{{DEFAULTSORT:Probability Theory}}

[[Category:Probability theory| ]]

[[id:Peluang (matematika)]]