Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 44 / 3 / f / Solution

After $k$ measurements the normal-normal conjugacy formulas areThe upper monitoring limit is $m_{ik}+z_{0.999}\sqrt{v_{ik}+{\sigma }_i^2}$. Its center moves toward the athlete's own mean and the population's influence diminishes. Its predictive width decreases toward $z_{0.999}{\sigma }_i$, since uncertainty about the mean vanishes but the next measurement's error remains. The absolute upper limit need not decrease monotonically: its center can move upward or downward with the observations. This is sequential normal predictive monitoring.