Given the two definitions:
arbitrary - based on random choice or personal whim, rather than any reason or system.
arbitrator/arbiter - an independent person or body officially appointed to settle a dispute.
What is the relationship between the two words? An arbitrator makes a choice when settling a dispute so there is obviously a link there, but it is not necessarily a choice that is random or on a whim. When we use the term arbitrary as in "an arbitrary length" the randomness portion of the definition is more important than the choice portion. Any idea how that came to be?