First a note: I am not a statistician. I studied maths at university (but opted out of every single stats class), and now find myself in a job where I'm doing stats.
My question is a little bit philosophical, and I'm sure that I have my brain in a twist by thinking too hard about the wrong things, but I'm struggling with the concept of a random variable.
Suppose a typist can type on average $\mu$ words per minute. If we take a sample of his (or her) typing we could approximate a normal distribution with mean $\mu $. I think that is pretty standard.
So we can say we expect the typist to type $\mu$ words per minute. OK, what if we replace the typist with another person? Are we to expect the second person to type at $\mu$ words per minute? Of course not, because we only have a sample for the first person and not the second, right? The second typist's mean could be completely different. That makes sense, but then what if I put the original typist back in and tie one hand behind his back. Obviously the expected value is different, because the typist clearly can't type as quickly. But then what conditions can we expect the typist's average to be $\mu$? It seems any perturbation in the typist's state changes the probability. But we never really defined the typist's state to begin with, the expected value was calculated from a sample. And obviously every time the typist types he or she is in a slightly different state, i.e. maybe has a headache, maybe fingers ache, maybe daydreaming, but we still count those in our sample. So it seems to me, that technically, if we can accept some small changes in state, then why can't we accept big changes in state? Or where is the cutoff? i.e. If I replace the typist, I have just as much right to say that the expected words per minute is still $\mu$.
I would like someone to explain why the above is wrong. Maybe it's an obvious answer.