= Solution
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.
\b[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.
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