The way I think of "random variable" is that it's an attempt to move toward a stateless world (a la functional programming) so we can reason about it.
What we actually observe in real life is a sample from the corresponding distribution. The "random variable" concept (a function over a sample space with a probability measure) is the stateless analogue that is more usable in abstract reasoning. In probability, we don't care about the specific values the variable actually ends up producing, but all the values it can produce and how likely each is.
In the observed world, it's unknown (pre-measurement) and appears to be random. In the mathematical world, it's a function over a sample space.
It's a mechanism to reason about randomness by (though the magic of probability) taking the randomness and state out.
What we actually observe in real life is a sample from the corresponding distribution. The "random variable" concept (a function over a sample space with a probability measure) is the stateless analogue that is more usable in abstract reasoning. In probability, we don't care about the specific values the variable actually ends up producing, but all the values it can produce and how likely each is.
In the observed world, it's unknown (pre-measurement) and appears to be random. In the mathematical world, it's a function over a sample space.
It's a mechanism to reason about randomness by (though the magic of probability) taking the randomness and state out.