What about the expectation of a `nearly' log-normal variable? [fact]
No one would have reacted badly to a distribution with an infinite fifth moment; nevertheless its exponential would also have an infinite expectation. Altogether, it used to be safe for scientists to believe that the universe is organized for simplicity and the conveniences of the mathematicians. But my fifty-year long study of complexity suggests otherwise. … For example, what can you deduce from knowing the average sales of a US software firm in 2003? It is the ratio with a numerator somewhat higher than the sales of Microsoft and a denominator that is meaningless because it is overly dependent on the smallest sales used in taking the average. What about the expectation of a `nearly' log-normal variable? It is so extremely sensitive to inevitable deviations from lognormality that it hardly matters. … The lesson for finance: divergences may or may not be significant, but must not be papered over in advance.
Donald MacKenzie, An Engine, Not a Camera, loc. 1525