below. In short, the answer is that a real valued random variable X is a real valued function. The domain of the function is a set of trials. So, for a trial w (usually written as lower case omega), X(w) is a real number.
Then the event for real number x
X <= x
is really shorthand notation for
{w|X(w) <= x}
So, typically we don't mention the w.
Moreover, typically for all but grad school mathematicians taking a course in "graduate probability" we don't mention that X is a function. Instead we just say something like, X is the number we get from running an experimental trial, one of all the numbers we "might have gotten" considering the probability distribution of X.
You are correct: You sense some mushy ground under the foundations of probability theory, and you are not nearly the first to so sense.
Long an answer was, "it works great in practice" which is doesn't make the mush any more firm.
Well, in 1933 A. Kolmogorov gave a solid mathematical foundation for probability theory. That's the usual foundations for advanced work in probability, statistics, and stochastic processes. My post
Some of the consequences are surprising, but I omit those. And we end up assuming that in all the universe all we ever see is just some one trial and don't say anything about the other trials but imagine about them a lot. That point may be hard to swallow.
IIRC, one line of argument is just that in probability there are lots of possibilities we just don't distinguish. E.g, maybe the police have long since concluded that nearly everyone driving a car with custom installed, hidden compartments is a drug dealer and then conclude that a person with such compartment is "likely" a drug dealer. Well, of course, actually, they might not be a drug dealer and have the car and its compartments for some other reason. So, the police are putting all owners of cars with hidden compartments in a box and refusing to distinguish them, insisting that they all be treated the same until there is evidence otherwise. It may be that more can be said. For now, make of such lines of thought what you will.
https://news.ycombinator.com/item?id=16854106
below. In short, the answer is that a real valued random variable X is a real valued function. The domain of the function is a set of trials. So, for a trial w (usually written as lower case omega), X(w) is a real number.
Then the event for real number x
X <= x
is really shorthand notation for
{w|X(w) <= x}
So, typically we don't mention the w.
Moreover, typically for all but grad school mathematicians taking a course in "graduate probability" we don't mention that X is a function. Instead we just say something like, X is the number we get from running an experimental trial, one of all the numbers we "might have gotten" considering the probability distribution of X.
You are correct: You sense some mushy ground under the foundations of probability theory, and you are not nearly the first to so sense.
Long an answer was, "it works great in practice" which is doesn't make the mush any more firm.
Well, in 1933 A. Kolmogorov gave a solid mathematical foundation for probability theory. That's the usual foundations for advanced work in probability, statistics, and stochastic processes. My post
https://news.ycombinator.com/item?id=16854106
here outlines that foundation.
Some of the consequences are surprising, but I omit those. And we end up assuming that in all the universe all we ever see is just some one trial and don't say anything about the other trials but imagine about them a lot. That point may be hard to swallow.
IIRC, one line of argument is just that in probability there are lots of possibilities we just don't distinguish. E.g, maybe the police have long since concluded that nearly everyone driving a car with custom installed, hidden compartments is a drug dealer and then conclude that a person with such compartment is "likely" a drug dealer. Well, of course, actually, they might not be a drug dealer and have the car and its compartments for some other reason. So, the police are putting all owners of cars with hidden compartments in a box and refusing to distinguish them, insisting that they all be treated the same until there is evidence otherwise. It may be that more can be said. For now, make of such lines of thought what you will.