I have a dataset that essentially amounts to a sample of K unique elements from an ordered sequence of length N. I believe that elements are more likely to be sampled from certain portions of the sequence (e.g. earlier elements are more likely to be selected), so my null hypothesis is that all elements are equally likely to be sampled. Equivalently, one could imagine an ordered sequence of K ones and (N-K) zeros, and I want to ask whether the ones appear to be uniformly distributed throughout the sequence or whether they vary in density along the sequence. What is a good statistical test to use for this question?
N is approximately 20000, and K is anywhere from 3000 to 10000.