Probability space

A probability space , or W-space for short , is a fundamental term from the mathematical sub-area of probability theory . It is a mathematical model for describing random experiments . The different possible outcomes of the experiment are combined into one set . Subsets of this result set can then, under certain conditions, be assigned numbers between 0 and 1, which are interpreted as probabilities .

The concept of probability space was introduced in the 1930s by the Russian mathematician Andrei Kolmogorow , who succeeded in axiomatizing the calculus of probability (see also: Kolmogorow axioms ).

definition

Formal definition

A probability space is a measure space whose measure is a probability measure . ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle P}$

In detail this means:

• ${\displaystyle \Omega }$is an arbitrary nonempty set called the result set . Their elements are called results .
• ${\displaystyle \Sigma }$is a σ-algebra over the basic set , i.e. a set consisting of subsets of , which contains and is closed with respect to the formation of complements and countable unions. The elements of hot events . The σ-algebra itself is also called the event system or event algebra . ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$
• ${\displaystyle P\colon \Sigma \to [0,1]}$is a probability measure, i.e. a set function that assigns numbers to events, such that is, for pairwise disjoint (i.e. mutually exclusive) events (3rd Kolmogorov axiom) and is (2nd Kolmogorov axiom). ${\displaystyle P(\emptyset )=0}$ ${\displaystyle P(A_{1}\cup A_{2}\cup \ldots )=P(A_{1})+P(A_{2})+\ldots }$ ${\displaystyle A_{1},A_{2},\dotsc }$ ${\displaystyle P(\Omega )=1}$

The measuring room is also called the event room . A probability space is therefore an event space on which a probability measure is also given. ${\displaystyle (\Omega ,\Sigma )}$

Specification of the definition

Modeling a wheel of fortune through a probability space: the set of possible outcomes is here . All subsets of are assigned their probabilities as a proportion of the angle of their sector in the full circle of the wheel. ${\displaystyle \Omega =\{1,2,3\}}$ ${\displaystyle \Omega }$

Specifically, the definition means that this model makes probability measurable as a purely axiomatically based construct (i.e. not empirically determined, as von Mises tried, and also not perceived subjectively). One of the key factors here is the idea of ​​constructing the set of all possible outcomes of the random experiment as mutually exclusive results. The example of the wheel of fortune makes this clear: When turning, the wheel can only stop in a single angular position to an imaginary zero position. As a result, however, only one of the three painted numbers 1, 2, 3 can be assigned to it, the wheel cannot stop in sector 1 and at the same time in sector 2. A mechanism prevents it from stopping exactly on the boundary between the two. This means that the simultaneous occurrence of two elementary events is excluded, they are disjoint. This justifies the transition from the general addition theorem to the special addition theorem , which corresponds to Kolmogorov's 3rd axiom : the probability of a union of countably many incompatible (i.e. mutually exclusive) events is equal to the sum of the probabilities of the individual events.

Classes of probability spaces

Discrete probability spaces

A probability space is called a discrete probability space if the result set is finite or countably infinite and the σ-algebra is the power set , that is . For some authors, when introducing the topic, the σ-algebra is omitted and the power set is tacitly assumed. Then only the tuple is called the discrete probability space. ${\displaystyle \Omega }$ ${\displaystyle \Sigma ={\mathcal {P}}(\Omega )}$ ${\displaystyle (\Omega ,P)}$

In some cases, probability spaces with an arbitrary basic set are also called discrete probability spaces if the probability measure almost certainly only takes on values ​​in a maximum that can be counted, i.e. it holds. ${\displaystyle \Omega }$ ${\displaystyle A}$ ${\displaystyle P(A)=1}$

Finite probability spaces

A finite probability space is a probability space whose basic set is finite and whose σ-algebra is the power set. Every finite probability space is a discrete probability space, accordingly the σ-algebra is partly omitted here as well. ${\displaystyle \Omega }$

Is special , provided with the Bernoulli distribution , that is , one speaks of a Bernoulli space . ${\displaystyle \Omega =\{0,1\}}$ ${\displaystyle P(\{0\})=p=1-P(\{1\})}$

Symmetric probability spaces

A symmetric probability space , also called Laplace's probability space or simply Laplace space (after Pierre-Simon Laplace ), consists of a finite basic set . The power set serves as the σ-algebra and the probability distribution is defined by a probability function as ${\displaystyle \Omega =\{\omega _{1},\dots ,\omega _{n}\}}$ ${\displaystyle \Sigma ={\mathcal {P}}(\Omega )}$

${\displaystyle P(\{\omega _{i}\})=f(i)={\frac {1}{n}}}$.

This corresponds exactly to the discrete uniform distribution . Symmetric probability spaces are always finite and therefore also discrete probability spaces. Accordingly, the σ-algebra is occasionally omitted here as well.

Other classes

Furthermore still exist

• induced probability spaces (image spaces of a random variable, provided with the distribution of the random variable as a probability measure)
• complete probability spaces ( complete measure spaces with a probability measure)
• Product spaces (see product of probability spaces )
• Filtered probability spaces : probability spaces, additionally provided with a filter .

Examples

Discrete probability space

An example of a discrete probability space is

• the result set of the natural numbers . Then every natural number is a result. ${\displaystyle \mathbb {N} =\{0,1,2,\dots \}}$
• As always, one chooses the power set as the event system for at most countably infinite sets . Then all subsets of the natural numbers are events. ${\displaystyle {\mathcal {P}}(\mathbb {N} )}$
• The Poisson distribution can be chosen as the probability measure, for example . It assigns the probability for a truly positive parameter to each set . ${\displaystyle P_{\lambda }}$ ${\displaystyle \{k\},k\in \mathbb {N} ,}$ ${\displaystyle P_{\lambda }(\{k\})={\frac {\lambda ^{k}}{k!}}\,\mathrm {e} ^{-\lambda }}$ ${\displaystyle \lambda }$

Then is a discrete probability space. ${\displaystyle (\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ),P_{\lambda })}$

Real probability space

An example of a real probability space is

• the result set of the non-negative real numbers . Then every non-negative real number is a result. ${\displaystyle \Omega =\mathbb {R} _{\geq 0}=[0,\infty )}$
• As an event system, Borel's σ-algebra on the real numbers, restricted to the non-negative real numbers . Then, for example, all closed, all half-open and all open intervals and their unions, cuts and complements are events. ${\displaystyle {\mathcal {B}}(\mathbb {R} )\cap [0,\infty )={\mathcal {B}}([0,\infty ))}$
• As a probability measure, for example, the exponential distribution . It shows the probability of every set in Borel's σ-algebra ${\displaystyle A}$

${\displaystyle P_{\mathrm {Exp} (\lambda )}(A)=\int _{A}\lambda \exp(-\lambda x)\mathrm {d} x}$

for a parameter too. ${\displaystyle \lambda >0}$

Then there is a probability space. ${\displaystyle ([0,\infty ),{\mathcal {B}}([0,\infty )),P_{\mathrm {Exp} (\lambda )})}$

Web links

• VV Sazonov: Probability space . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Probability Space . In: MathWorld (English).

literature

• 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 .
• Norbert Henze : Stochastics for beginners. 10th edition. Springer Spektrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 37, p. 180, p. 283.
• 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 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ↑ Ulrich Krengel : Introduction to Probability Theory and Statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , p. 3 , doi : 10.1007 / 978-3-663-09885-0 .
2. ↑ David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 63 , doi : 10.1007 / b137972 .
3. ↑ Ehrhard Behrends: Elementary Stochastics . A learning book - co-developed by students. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-8348-1939-0 , pp. 40 , doi : 10.1007 / 978-3-8348-2331-1 .
4. ^ Schmidt: Measure and probability. 2011, p. 198.
5. ^ Georgii: Stochastics. 2009, p. 27.