Product model (stochastics)

The title of this article is ambiguous. For the product of statistical models, see Product Model (Statistics) .

A product model , partly product experiment is a term from the stochastics , a partial area of mathematics . A product model formalizes the idea that an attempt, for example a coin toss, can be carried out as often as desired, independently of one another. In this context one speaks of the product of probability spaces .

definition

Let a finite or countably infinite index set be given , i.e. or and for each a probability space is given. Here is the result set , the event system , a σ-algebra , and a probability measure . Then the probability space is called ${\displaystyle I}$ ${\displaystyle I=\{1,\dots ,n\}}$ ${\displaystyle I=\mathbb {N} }$ ${\displaystyle i\in I}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i},P_{i})}$ ${\displaystyle \Omega _{i}}$ ${\displaystyle {\mathcal {A}}_{i}}$ ${\displaystyle P_{i}}$

${\displaystyle \left(\prod _{i=1}^{|I|}\Omega _{i},\bigotimes _{i=1}^{|I|}{\mathcal {A}}_{i},\bigotimes _{i=1}^{|I|}P_{i}\right)}$

the product of the probability spaces or simply a product experiment or product model. Here is ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i},P_{i})}$

${\displaystyle \prod _{i=1}^{|I|}\Omega _{i}}$

the Cartesian product of the result sets,

${\displaystyle \bigotimes _{i=1}^{|I|}{\mathcal {A}}_{i}}$

the product σ-algebra of the σ-algebras and ${\displaystyle {\mathcal {A}}_{i}}$

${\displaystyle \bigotimes _{i=1}^{|I|}P_{i}}$

the product measure of the probability measures . ${\displaystyle P_{i}}$

Examples

It be and for each is and . Each of the individual experiments is therefore a fair coin toss. The fivefold product experiment is then the fivefold independent tossing of a coin, the product space is then , where the probability measure is defined by the equal distribution on , i.e. for all . ${\displaystyle I=\{1,\dots ,5\}}$ ${\displaystyle i\in I}$ ${\displaystyle \Omega _{i}=\{0,1\},\,{\mathcal {A}}_{i}={\mathcal {P}}(\{0,1\})}$ ${\displaystyle P_{i}(\{1\})={\tfrac {1}{2}}=P_{i}(\{0\})}$ ${\displaystyle \left(\{0,1\}^{5},{\mathcal {P}}(\{0,1\}^{5}),P\right)}$ ${\displaystyle \Omega =\{0,1\}^{5}}$ ${\displaystyle P(\{\omega \})={\tfrac {1}{2^{5}}}}$ ${\displaystyle \omega \in \Omega }$

Features and remarks

• If they are all the same, you also write for the product area. ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i},P_{i})}$ ${\displaystyle (\Omega _{i}^{I},{\mathcal {A}}_{i}^{\otimes I},P_{i}^{\otimes I})}$
• If the projection is from the -th component of the product space , then the distribution of is also called a marginal distribution or an edge distribution . ${\displaystyle X_{i}}$ ${\displaystyle i}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle X_{i}}$

existence

Problems in the construction of a general product model are primarily the product dimensions. In the case of a finite number of repetitions, Carathéodory's measure extension theorem guarantees existence. In addition, there are existential statements for countably infinite products of finite probability spaces. Only the Andersen-Jessen theorem solves the general case for countable or uncountable many products of probability measures.

use

Product experiments are widely used in statistics and stochastics. For example, they form the basis for the definition of some probability measures that can be defined as waiting time distributions , such as the geometric distribution . In statistics, they enable the modeling of situations in which samples are successively enlarged in order to be able to make statements about the quality of estimators, for example.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Ulrich Krengel: Introduction to probability theory and statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , doi : 10.1007 / 978-3-663-09885-0 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .