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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# <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==