Is Sample Space an Open or Closed Set?

[woundedtiger4]

Hi all,

is sample space an open set or a closed set?

Thanks in advance

 

[chiro]

Hey woundedtiger4.

What do you mean by sample space exactly?

If you have a set with a finite number of elements, it has to be closed by default but if you are talking about some arbitrary region, then it may be different.

The reason I say the above is that if you are talking about a sample in statistics, a sample is always finite that is drawn from either an infinite population or a finite one (like for example when you have populations like the people in an entire country or state which is used in survey design as an example).

 

[chiro]

Hey woundedtiger4.

What do you mean by sample space exactly?

If you have a set with a finite number of elements, it has to be closed by default but if you are talking about some arbitrary region, then it may be different.

The reason I say the above is that if you are talking about a sample in statistics, a sample is always finite that is drawn from either an infinite population or a finite one (like for example when you have populations like the people in an entire country or state which is used in survey design as an example).

 

[ImaLooser]

Hi all,

is sample space an open set or a closed set?

Thanks in advance

Whatever you like. Any set will do, as far as I know.

 

[chiro]

Is the sample a statistical sample or something else?

 

[mathman]

A sample space is not required to have a topology, so open or closed is besides the point.

 

[woundedtiger4]

A sample space is not required to have a topology, so open or closed is besides the point.

why?

Edited: I mean why it doesn't require topology?

 

[Stephen Tashi]

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.

 

[ImaLooser]

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.

Your sample space can be a housecat, an anchovy pizza, and a Mohair-covered Caddillac. So no, you don't need a topology.

 

[woundedtiger4]

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.

excellent explanation... thanks