Unit vector and Probability space: Difference between pages
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[[File:Probability-measure.svg|thumb|Modelling a wheel of fortune using probability space]] | |||
'''Probability space''' is a [[mathematical model]] used to describe [[scientific]] [[experiments]] A probability space consists of three parts: | |||
# A [[sample space]] which lists all possible outcomes | |||
# A [[set]] of events. Each event associates zero or more outcomes | |||
# A [[Function (mathematics)|function]] that assigns probabilities to each event | |||
An ''outcome'' is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex ''events'' are used to characterize groups of outcomes. The collection of all such events is a ''[[σ-algebra]]'' <math>\scriptstyle \mathcal{F}</math>. Finally, there is a need to specify each event's likelihood of happening. This is done using the ''[[probability measure]]'' function, ''P''. | |||
Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ''ω'', from the sample space Ω. All the events in <math>\scriptstyle \mathcal{F}</math> that contain the selected outcome ''ω'' (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function ''P''. | |||
The prominent Soviet mathematician [[Andrey Kolmogorov]] introduced the notion of probability space, together with other [[axioms of probability]], in the 1930s. | |||
[[Category:Mathematics]] | |||
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Latest revision as of 19:23, 8 November 2016
Probability space is a mathematical model used to describe scientific experiments A probability space consists of three parts:
- A sample space which lists all possible outcomes
- A set of events. Each event associates zero or more outcomes
- A function that assigns probabilities to each event
An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex events are used to characterize groups of outcomes. The collection of all such events is a σ-algebra [math]\displaystyle{ \scriptstyle \mathcal{F} }[/math]. Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, P.
Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ω, from the sample space Ω. All the events in [math]\displaystyle{ \scriptstyle \mathcal{F} }[/math] that contain the selected outcome ω (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function P.
The prominent Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s.