Result (stochastics)

A result is a term from the basics of stochastics . Many different names, including himself in the literature random result , basic event , atomic event , element of a probability space , feature or natural disaster . The designation as a natural event is ambiguous, see section Natural events . Results can be introduced in two ways: either as an element of the result set in a probability space or as a possible outcome of a modeled random experiment . ${\ displaystyle \ Omega}$

definition

There are two approaches to defining results:

If one comes axiomatically from the definition of a probability space , then each element of the result set is called a result ${\ displaystyle (\ Omega, \ Sigma, P)}$ ${\ displaystyle \ Omega}$

If one starts from a random experiment and wants to model a corresponding probability space based on this, then every possible outcome of the random experiment is called a result. The results are then summarized in the result set.

Mostly results are indicated with . ${\ displaystyle \ omega}$

Examples

Examples of results as the outcome of a random experiment are:

The throw of a dice is to be modeled. Possible outputs are the numbers from 1 to 6. So are the results . ${\ displaystyle \ omega _ {1} = 1, \ omega _ {2} = 2, \ omega _ {3} = 3, \ omega _ {4} = 4, \ omega _ {5} = 5, \ omega _ {6} = 6}$
The service life of an electrical component is to be modeled. Since it can break at any time after the start of the observation, the results are of the form for and . ${\ displaystyle \ omega = x}$ ${\ displaystyle x \ geq 0}$ ${\ displaystyle x \ in \ mathbb {R}}$
Results are not always structured that simply. For example, if one considers possible charge distributions in a crystal lattice and wants to make statements about possible transitions, one result can be a node-weighted graph with node weights 0, 1 or −1.

Example of results as elements of the result set is:

If one looks at the probability space with an arbitrary probability measure , then every natural number is a result because it is an element of . ${\ displaystyle (\ mathbb {N}, {\ mathcal {P}} (\ mathbb {N}), P)}$ ${\ displaystyle P}$ ${\ displaystyle \ mathbb {N}}$

Role in modeling

Results are the smallest units in the definition of a stochastic model. They are not yet assigned a probability, but are combined to form the result set .

On the result set you now define the sets to which a probability is to be assigned, the events . These in turn are collected in the event system , a σ-algebra .

The event system forms the counterpart to the definition set in analysis. A probability can only be assigned to the elements of the event system.

A triple of result set , event system and probability measure is also called a probability space and forms the basis for further investigations. ${\ displaystyle \ Omega}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle P}$

Results and Events

Results and events are easy to confuse.

Outcomes are elements of a crowd; they cannot be given a probability. For example, it is a result of rolling the dice. ${\ displaystyle 6}$
Events are subsets of the result set. They therefore contain results as elements. So when rolling the dice there is an event but not a result. Conversely, if there is a result, it does not necessarily have to be an event. Events can contain any number of elements, such as . ${\ displaystyle \ {6 \}}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ {\ omega \}}$ ${\ displaystyle \ {1,2 \}}$

Acts of God

The term “act of God” is not clearly used in the literature. Sometimes it denotes a result , then the name “event” is misleading because results and events are different. In some cases, even with a discrete result set, it denotes an event with one element, i.e. of the form . ${\ displaystyle \ omega}$ ${\ displaystyle \ {\ omega \}}$

term

The term "elementary event" for the elements of the probability space goes back to Kolmogorow ; Although this also made a distinction between the elements of the result set and their one-element subsets, it did not introduce a separate name for the latter. In contrast, more recent literature uses the terms “result” or “output”. “Event” is clearly understood as a set consisting of results.

literature

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 .

Individual evidence

↑ ^a ^b ^c Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 195 , doi : 10.1007 / 978-3-642-21026-6 .
↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 3 , doi : 10.1007 / 978-3-642-17261-8 .
↑ Hesse: Applied probability theory. 2003, p. 19.
↑ Krengel: Introduction to Probability Theory and Statistics. 2005, p. 3.
↑ ^a ^b Norbert Henze : Stochastics for beginners . An introduction to the fascinating world of chance. 10th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 1 , doi : 10.1007 / 978-3-658-03077-3 .

Web links

Elementary event on math online

[Schmidt195-1] Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 195 , doi : 10.1007 / 978-3-642-21026-6 .

[2] Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 3 , doi : 10.1007 / 978-3-642-17261-8 .

[3] Hesse: Applied probability theory. 2003, p. 19.

[4] Krengel: Introduction to Probability Theory and Statistics. 2005, p. 3.

[Henze1-5] Norbert Henze : Stochastics for beginners . An introduction to the fascinating world of chance. 10th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 1 , doi : 10.1007 / 978-3-658-03077-3 .