Probability theory: Difference between revisions

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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==