Probability theory: Difference between revisions

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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 dicedie 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 dice rolls. These collections are called ''events''. In this case, {1,3,5} is the event that the dicedie 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 dicedie, the assignment of a number to 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.

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# <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 CDF <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 CDF back again,. thenIn this case, the random variable ''X'' is said to have a {{em|[[probability density function]]}} ({{em|PDF}}) 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

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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 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 CDF converges<math>F_1,F_2,\dots\,</math> convergeconverges to the CDF <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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