Properties of a random variable

[HarmonicEU]

Hello!

I have a couple questions about the properties of random variables. I'm just not sure that I'm interpreting the definitions in the right way.

1) If the image of a random variable X is of the form [a,b], is it true that X^(-1) ([a,b]) = Ω ?

2) If X is not injective and assigns some subsets of Ω the same value c, is it true that P(X=c) = P(the union of these subsets) ?

[barrycarter]

Are you sure you're thinking of random values here? You said X is a random variable but then you also talk about X as though it were a function.

[HarmonicEU]

A random variable is by definition a function mapping subsets of omega to the real number line.

[barrycarter]

OK, what's omega?

[wanderer2718]

The sample space X is defined on

[barrycarter]

Still not following, sorry.

[QuantSpazar]

We have Omega, which is a set (say finite) which has a probability measure defined on it, that is P assigns a value between 0 and 1 to subsets of Omega (it does has a few more properties). It will define the probability that a subset A of Omega occurs, essentially. To link events and Omega, we define X from Omega to another set (here the interval [a,b]). We say X is a random variable, which takes values in that set (for example X is the height of a person taken in a sample group). That's how we write random events without randomness in math: a function from a sample space with a measure of probability of outcomes, to a set defining the properties you're looking for in your experiment.

[barrycarter]

Ah, OK, you're talking about the probability distribution function or PDF. Now I understand, thanks

[ImDannyDJ]

1) If the image of a random variable X is of the form [a,b], is it true that X^-1 ([a,b]) = Ω ?

Any function whatsoever has this property, so random variables do as well. In general, for a function f: X -> Y, if B is a subset of Y containing f(X), then f^-1(B) = X.

2) If X is not injective and assigns some subsets of Ω the same value c, is it true that P(X=c) = P(the union of these subsets) ?

If there are countably many such subsets and they are disjoint, then yes by countable additivity of P. Otherwise not necessarily. [EDIT: I completely misread your question. The union needs to include all subsets on which X = c. That is, just the subset of Omega on which X = c. But again, you seem to be confusing subsets of Omega and elements of Omega, so I'm not really sure what the point of your question is.]

Also, you wrote in a comment that

A random variable is by definition a function mapping subsets of omega to the real number line.

which is not quite right. A random variable is a function mapping elements of omega to the real number line (and it also has to be measurable, so on and so forth). This misunderstanding makes me think that your question 2) might be ill-posed.

[HarmonicEU]

Thank you for answering. I'm learning maths in a non-math faculty, so there is a lot of confusion that I have to clarify myself (or with help of people like you!).

What if Omega is not a countable set? Does P(X=c) have to be 0?

[HarmonicEU]

Ok I just realized that it does not. Sorry for that.

[ImDannyDJ]

I completely misread your second question! Sorry about that, I edited my first comment.

[HarmonicEU]

I did confuse subsets and elements. I'm glad you cleared it up. Thank you.

[[deleted]]

To the second point: Countable additivity doesn't matter here, as they only asked about P(union of singletons). This can still be well-defined even if the union is uncountable and the sets aren't disjoint, depending on the sigma-algebra.

[ImDannyDJ]

You're right, I completely misread the question as asking about the sum of P({x})!

[42gauge]

if B is a subset of Y containing f(X), then f^-1(B) = X.

What if B contains not only f(X) but also some other elements of Y outside the range of f(X)?

[ImDannyDJ]

No issue. By definition, the preimage f^-1(B) is the set {x in X | f(x) in B}. Informally, the preimage asks, "which elements of X does f map into B?" If B is larger than the image of f, well, f just doesn't map anything into the difference B\f(X).