This is a very simple exercise designed to clarify confusion about the roles of past, present, and future in stopping rules. Let {X n ; n ≥ 1} be a sequence of IID binary rv’s , each with the pmf...

This is a very simple exercise designed to clarify confusion about the roles of past, present, and future in stopping rules. Let {X_n; n ≥ 1} be a sequence of IID binary rv’s , each with the pmf pX(1) = 1/2, pX(0) = 1/2. Let J be a positive integer-valued rv that takes on the sample value n of the first trial for which X_n
= 1. That is, for each n ≥ 1,