probability of rank for n iid random variables Denote the $i$th order statistic as $X_{(i)}$, then what is $\Pr(X_i = X_{(j)})$ for some $i, j = 1,...,n$ and iid $X_1, ..., X_n$?
I think this is what rank tests are based on but I'm stuck on making any progress.
 A: If you have $n$ real-valued continuous random variables which are identically and independently distributed, then that means there is no bias as to which variable is realised as which order statistic.
$$\begin{align}\mathsf P(X_i=X_{(j)}) & = \mathsf P(X_1=X_{(j)}) \\[1ex] & = \mathsf P(X_2=X_{(j)}) \\[1ex] & \vdots \\[1ex] & = \mathsf P(X_n=X_{(j)}) \\[1ex] & = \frac{1}{n}\end{align}$$
Note that this doesn't quite hold if there is a non-zero measure of 'ties' (such as for discrete distributions).   In such a case the events have equal probability, but are not mutually exclusive.
A: This might help you see it more easily:
Imagine that you write the order statistics on balls and put them in a bag, mix them up and draw them out one at a time in order to pair the observation numbers (the "$i$" in your notation) with those values. So label the first ball drawn "observation 1", the second "observation 2" and so on.
What's the probability that the ball with $j$th order statistic is drawn first? Drawn 14th? Drawn last?
