This work studies variations of optimal prediction problems introduced in Shiryaev, Zhou and Xu (2008) and Du Toit
and Peskir (2009) to a randomized terminal-time set up and different families of utility
measures. We analyze criteria under which the existence of optimal stopping rules is guaranteed and characterize the value of the
stopping problem as the solution to a family of free boundary problems. Also, we
introduce a numerical technique in order to build approximations of stopping boundaries for
fixed terminal time problems and suggest previously reported stopping rules extend
to certain generalizations of measures.