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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{{Main|Discrete probability distribution}}

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

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

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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}}.

===Continuous probability distributions===

{{Main|Continuous probability distribution}}

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

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

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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==