Pdf and pmf as random variables?

[Rasalhague]

If the set of real numbers is considered as a sample space with the Borel sigma algebra for its events, and also as an observation space with the same sigma algebra, is a pdf or pmf a kind of random variable? That is, are they measurable functions?

 

[micromass]

Hi Rasalhague!

If the set of real numbers is considered as a sample space with the Borel sigma algebra for its events, and also as an observation space with the same sigma algebra, is a pdf or pmf a kind of random variable? That is, are they measurable functions?

Yes, a pdf is always measurable, it is even integrable. In fact, a pdf is defined to be integrable.
A pmf is certainly measurable since its domain is countable.

Almost all functions you will ever encounter in probability theory will be measurable, so this is (luckily) no exception to that rule.

 

[Rasalhague]

Hello, again, micromass!

A pmf is certainly measurable since its domain is countable.

Is a pmf also measurable when, as here, it's defined with an uncountable domain, namely the real numbers?

 

[disregardthat]

Is a pmf also measurable when, as here, it's defined with an uncountable domain, namely the real numbers?

Are you sure about that?

 

[micromass]

Hello, again, micromass!



Is a pmf also measurable when, as here, it's defined with an uncountable domain, namely the real numbers?

Yes, because there are at most countable non-boring numbers. That is, most of the numbers in the uncountable domain are being sent to 0, while only a countable number of them are interesting. This means that it's measurable.

 

[Rasalhague]

Aha, I think I see why it has to be! A pmf, f_X, has (finitely or infinitely) countable range (because it has only countably many nonzero, i.e. non-boring elements), so every subset is a countable union of singletons, which are elements of the Borel algebra on R being complements of pairs of open sets. The pre-image of every subset not containing zero is a subset of the range of X, also a countable union of singletons, because X is discrete. The pre-image of every subset containing zero is a countable union of singletons and the open intervals between them, together with the open interval before the first element of the range of X and the open interval after the last. So these pre-images are also elements of the Borel algebra. So f_X is measurable.

 

[Rasalhague]

Are you sure about that?

Sure that the Wikipedia article I linked to defines the pmf on R? Yes, unless it's been changed recently, it's in the 2nd sentence of "Formal definition" and reiterated in the sentence immediately after that.

Here's another source which defines the pmf on R.

 

[disregardthat]

Sure that the Wikipedia article I linked to defines the pmf on R? Yes, unless it's been changed recently, it's in the 2nd sentence of "Formal definition" and reiterated in the sentence immediately after that.

Here's another source which defines the pmf on R.

You are right, I accidentally thought you were talking about the random variable.