The blimpy shape of heady- s and taily- s bit strings

This report documents the unimodal shape of two discrete functions related to the number of binary sequences under a certain bit-scoring scheme along with some attempts to prove the unimodality rigorously. While a completely general proof is not yet in hand, we state several monotonicity properties for certain related, novel discrete distributions which imply the main results and which we conjecture hold true in general. Connections to sequential stopping times under required conditioning events are discussed and evaluated for their ability to establish the results. The "blimpy" shape of the number of bit strings with a given score as a function of the score is explained and accurately approximated with a four-parameter model.