The question whether a time average over a long period of a random quantity in one system and an average over multiple systems over a short period are the same is the question for ergodicity [1]. If both are the same, the random process or system is called ergodic.

The answer is: If the measurement period for the average over multiple systems is long enough that each random variable has visited a representative sample of states which make up the majority of contributions to the mean value, then the system is ergodic.

[1] https://en.wikipedia.org/wiki/Ergodicity