  We adopted survival analysis for the viewing durations of massive open online
courses. The hazard function of empirical duration data is dominated by a
bathtub curve and has the Lindy effect in its tail. To understand the
evolutionary mechanisms underlying these features, we categorized learners into
two classes due to their different distributions of viewing durations, namely
lognormal distribution and power law with exponential cutoff. Two random
differential equations are provided to describe the growth patterns of viewing
durations for the two classes respectively. The expected duration change rate
of the learners featured by lognormal distribution is supposed to be dependent
on their past duration, and that of the rest learners is supposed to be
inversely proportional to time. Solutions to the equations predict the features
of viewing duration distributions, and those of the hazard function. The
equations also reveal the feature of memory and that of memorylessness for the
viewing behaviors of the two classes respectively.
