Independent and identically distributed random variables

Also refered to by the shorthand: i.i.d.

This is a mathematical idealization that captures the idea of a repeatable experiment. The intuition is that each experiment will give rise to the same distribution of possible outcomes and that moreover, knowledge of the actual outcomes for any set of experiments will not help you in the slightest in predicting the outcome of any other collection of experiments.

A set of random variables are i.i.d., if each one has the same [probability distribution] and furthermore, the random variables are all [independent] of each other. This independence has to go beyond mere pairwise independence. What is desired is that all partitions of the set of random variables has the property that each partition is independent of all others.