Unfortunately, this is a well-established mathematical term that’s used universally throughout probability theory. Changing notation to match intuition is not feasible; we must instead change intuition to match notation.
Technically speaking, a random variable is just a measurable function X:Ω→E, where Ω is the underlying sample space and E is some measurable space. Indeed, even if the function X is constant, it is technically speaking still a random variable (in this case it’s called a deterministic random variable).
The problem is that, in practice, people don’t really pay too much attention to precisely what the sample space is, especially if they don’t have some specific reason to care about measure theory. They instead often want to talk about the probability distribution, and the task of figuring out formally and rigorously why this is all well-defined is often omitted. Luckily, the Kolmogorov extension theorem and related results usually allow you to pick a “large enough” sample space that carries all the content you need for your math work.
stochastic variable is certainly less common, but googling it only returns the right thing. it seems like it’d be a valid replacement and I agree it could reduce a common confusion.
“Random variable” is never defined. I though stochastic variable is just a synonym for random variable. I have seen posts where random variable is always written as r.v. and that helps a bit.
From Wikipedia: “In probability theory, the sample space (also called sample description space,[1]possibility space,[2] or outcome space[3]) of an experiment or random trial is the set of all possible outcomes or results of that experiment.
what is a measurable space?
“he function X is constant,” you mean its just one outcome like a die that always lands on one side?
I’m not sure if clarifying this is most useful for the purpose of understanding this post specifically, but for what it’s worth: A measurable space is a set together with a set of subsets that are called “measurable”. Those measurable sets are the sets to which we can then assign probabilities once we have a probability measure P (which in the post we assume to be derived from a density p, see my other comment under your original comment).
“the function X is constant,” you mean its just one outcome like a die that always lands on one side?
I think that’s what the commenter you replied to means, yes. (They don’t seem to be active anymore)
what makes a function measurable?
This is another technicality that might not be too useful to think about for the purpose of this post. A function is measurable if the preimages of all measurable sets are measurable. I.e.: f:X→Z, for two measurable spaces X and Z, is measurable, if f−1(A)⊆X is measurable for all measurable A⊆Z. For practical purposes, you can think of continuous functions or, in the discrete case, just any functions.
Unfortunately, this is a well-established mathematical term that’s used universally throughout probability theory. Changing notation to match intuition is not feasible; we must instead change intuition to match notation.
Technically speaking, a random variable is just a measurable function X:Ω→E, where Ω is the underlying sample space and E is some measurable space. Indeed, even if the function X is constant, it is technically speaking still a random variable (in this case it’s called a deterministic random variable).
The problem is that, in practice, people don’t really pay too much attention to precisely what the sample space is, especially if they don’t have some specific reason to care about measure theory. They instead often want to talk about the probability distribution, and the task of figuring out formally and rigorously why this is all well-defined is often omitted. Luckily, the Kolmogorov extension theorem and related results usually allow you to pick a “large enough” sample space that carries all the content you need for your math work.
https://en.wikipedia.org/wiki/Random_variable
stochastic variable is certainly less common, but googling it only returns the right thing. it seems like it’d be a valid replacement and I agree it could reduce a common confusion.
“Random variable” is never defined. I though stochastic variable is just a synonym for random variable. I have seen posts where random variable is always written as r.v. and that helps a bit.
From Wikipedia: “In probability theory, the sample space (also called sample description space,[1] possibility space,[2] or outcome space[3]) of an experiment or random trial is the set of all possible outcomes or results of that experiment.
what is a measurable space?
“he function X is constant,” you mean its just one outcome like a die that always lands on one side?
what makes a function measurable?
I’m not sure if clarifying this is most useful for the purpose of understanding this post specifically, but for what it’s worth: A measurable space is a set together with a set of subsets that are called “measurable”. Those measurable sets are the sets to which we can then assign probabilities once we have a probability measure P (which in the post we assume to be derived from a density p, see my other comment under your original comment).
I think that’s what the commenter you replied to means, yes. (They don’t seem to be active anymore)
This is another technicality that might not be too useful to think about for the purpose of this post. A function is measurable if the preimages of all measurable sets are measurable. I.e.: f:X→Z, for two measurable spaces X and Z, is measurable, if f−1(A)⊆X is measurable for all measurable A⊆Z. For practical purposes, you can think of continuous functions or, in the discrete case, just any functions.