Result space

As a result, space , result set , result quantity , Omega amount or sample space is called the mathematical branch of stochastics , the amount of all possible results of a random experiment . To describe such an experiment with the help of a probability space , certain subsets of the result space , the events , are assigned probabilities. ${\ displaystyle \ Omega}$

To set up a suitable sample space in multi-stage random experiments, can sometimes as a simple tool decision tree can be used.

Examples

When rolling a die , the result space is: ${\ displaystyle \ Omega = \ {1,2,3,4,5,6 \}}$

With a simple coin toss, the result space is:

{\ displaystyle \ Omega = \ {K, Z \}; (K = {\ text {head}}, Z = {\ text {number}})}

When tossing two distinguishable coins at the same time, the result space is:, where the large coins are represented by and the small coins by .

{\ displaystyle \ Omega = \ {Kk, Kz, Zz, kZ \}}

{\ displaystyle (K = {\ text {head}}, Z = {\ text {number}})}

{\ displaystyle (k = {\ text {head}}, z = {\ text {number}})}

It is entirely possible that there are two or more reasonable result spaces for a random experiment. For example, if you look at the random experiment of drawing a card from a deck of cards , the result set can include the card values (ace, 2, 3, ...) or the suit values (clubs, spades, hearts, diamonds). However, a full listing of the results would take into account both card value and suit. A corresponding result set can be generated as a Cartesian product of the two previous result sets.

meaning

To calculate the probability of discrete events according to Laplace, it is essential to know the thickness of the result space . Result spaces also occur with probability spaces. A probability space builds on a result space, but defines a set of “interesting events”, the event algebra , on which the probability measure is defined. For a more explicit presentation in context and with an example see probability theory . ${\ displaystyle (\ Omega, \ Sigma, P)}$ ${\ displaystyle \ Omega}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle P}$

Definition of terms: event space - result space

The concept of the result space is the analogue of the event space in inductive statistics .

In the literature, a careful distinction is not always made between the terms event system , event space (in the sense of the measurement space) and result space . Therefore it happens that the result space is called an event space.

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 .
Rainer Schlittgen : Introduction to Statistics: Analysis and Modeling of Data. 9th edition, Oldenbourg, Munich Vienna 2000, ISBN 3-486-25465-0

Individual evidence

^ Georgii: Stochastics. 2009, p. 8.

[1] Georgii: Stochastics. 2009, p. 8.