Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 2 / h

The posterior distribution has a long right tail. Its posterior mean is pulled upward by relatively rare large endpoints and is sensitive to the artificial upper cutoff; its untruncated posterior mean does not exist. The median is a central quantile that stays finite as the cutoff is removed and is much less sensitive to extreme draws. It is also the Bayes estimator under absolute-error loss, whereas a mean corresponds to squared-error loss when the relevant expectation exists.

The median better represents the typical posterior endpoint in this strongly right-skewed distribution. It need not be preferable for every decision problem, since the loss function matters. For comparison, exact summation of the truncated kernel at cutoff $5000$ gives median $195$ and mean about $397.67$; the displayed simulation summaries are Monte Carlo estimates, not exact posterior calculations.