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woundedtiger4
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Hi all,
is sample space an open set or a closed set?
Thanks in advance
is sample space an open set or a closed set?
Thanks in advance
woundedtiger4 said:Hi all,
is sample space an open set or a closed set?
Thanks in advance
mathman said:A sample space is not required to have a topology, so open or closed is besides the point.
Stephen Tashi said:In a topology, the set that is the whole space is both open and closed. So if you define "sample space" to be the set of all points under consideration, it is both open and closed in any topology that you define on it.
Whether you must have a toplogy to do probability theory is an interesting question. To view probability from the point of view of masure theory, you must have a "sigma algebra" of sets. Some of the axioms for a sigma algebra are very similar to those for a toplogy, but according to this discussion http://mathoverflow.net/questions/70137/sigma-algebra-that-is-not-a-topology , a sigma algebra need not be a toplogy. From that point of view, you can do probability theory without a topology.
If you are taking a course that focuses on applications of probability and doing summations or integrations of functions defined on real numbers then your course assumes the "usual" topology for 1 or n-dimensional euclidean space.
Stephen Tashi said:In a topology, the set that is the whole space is both open and closed. So if you define "sample space" to be the set of all points under consideration, it is both open and closed in any topology that you define on it.
Whether you must have a toplogy to do probability theory is an interesting question. To view probability from the point of view of masure theory, you must have a "sigma algebra" of sets. Some of the axioms for a sigma algebra are very similar to those for a toplogy, but according to this discussion http://mathoverflow.net/questions/70137/sigma-algebra-that-is-not-a-topology , a sigma algebra need not be a toplogy. From that point of view, you can do probability theory without a topology.
If you are taking a course that focuses on applications of probability and doing summations or integrations of functions defined on real numbers then your course assumes the "usual" topology for 1 or n-dimensional euclidean space.
A sample space is a set of all possible outcomes of an experiment or event.
The answer to this question depends on the specific context. In some cases, a sample space may be considered an open set, while in others it may be considered a closed set. It is important to define the sample space clearly in order to determine its properties.
Sample space is an important concept in probability theory. It is used to define the set of all possible outcomes of an experiment, which is then used to calculate the probability of a specific outcome occurring.
Yes, a sample space can be infinite. For example, the sample space for rolling a six-sided die is infinite, as there is no limit to the number of times the die can be rolled.
Sample space and event space are related concepts, but they are not the same. Event space is a subset of the sample space, consisting of specific outcomes or combinations of outcomes that are of interest in a given experiment or event.