identically distributed random variables

Two random variables are identically distributed when they follow the same probabilistic recipe, even if they are produced by different experiments or take different actual values on a given day. Two fair dice, rolled separately, are identically distributed: each gives 1 through 6 with probability 1/6. On one throw the first might show 3 and the second 5, but their distributions, the full pattern of how likely each value is, are exactly the same.

Precisely, X and Y are identically distributed if they have the same cdf: F_X(x) = F_Y(x) for every x. Equivalently they have the same pmf (discrete) or the same pdf (continuous). This says nothing about whether they tend to move together; it is purely a statement that, taken one at a time, they obey the identical law. Note the contrast with being equal: X = Y means they always produce the same number on every outcome, a far stronger condition than merely sharing a distribution.

The idea earns its keep in statistics. A sample is usually modelled as independent and identically distributed (iid): each observation is drawn from the same population (identically distributed) and does not influence the others (independent). That shared distribution is what we are trying to learn about from the data, and the workhorse theorems, the law of large numbers and the central limit theorem, are stated for iid sequences. Honest caveat: identically distributed and independent are separate properties; variables can share a distribution yet be strongly dependent, like X and -X for a symmetric X.