Probability theory: Difference between revisions

Revision as of 13:25, 27 February 2023

Revision as of 14:25, 27 February 2023

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

Whereas the ''pdfPDF'' exists only for continuous random variables, the ''CDF'' 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.

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The ''[[wikt:raison d'être|raison d'être]]'' 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 CDF, and vice versa. The measure corresponding to a CDF is said to be {{em|induced}} by the CDF. 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