Say each doctor's patients had a common constant death rate for that doctor, not a crazy assumption, and those constant death rates across doctors followed a gamma distribution (not hard, because I could fit my ass with a gamma), then the distribution of each doctor's patients' life expectancies would be exponential and the distribution of life expectancies of the whole group would be Pareto.

Exponential is completely positively skewed and thin-tailed, and Pareto is roughly the same shape but very fat-tailed.