Introduction to Probability, Second Edition, 2nd Edition

5.7    Symmetry of i.i.d. continuous r.v.s

Continuous r.v.s that are independent and identically distributed have an important symmetry property: all possible orderings are equally likely. Intuitively, this is because if all we are told is that $X_1,\dots ,X_n$ are i.i.d., then they are interchangeable, in the sense that we have been given no information that distinguishes one $X_i$ from another $X_j$. This is reminiscent of the fact that it is common for someone to say “Do you want the good news first or the bad news first?” but rare for someone to say “I have two pieces of news. Which do you want to hear first?”, since in the latter case no distinguishing information has been provided for the two pieces of news.

Proposition 5.7.1. Let $X_1,\dots ,X_n$ be i.i.d. ...