Properties of Pyramids and their Pharoahs

Question

(A) It's been noted that the volumes of pyramids tend to be a linear function of the duration of the reign of the pharoah originally buried in each. Assume that the duration of the reign of pharoahs is a random number over the range of 5 - 50 years, and that all pyramids are geometrically similar. Would you expect the average height of the pyramids (over many pharoahs) to be less than, the same as, or greater than, the height of the pyramid of the average-duration-of-reign pharoah (27.5 years)? Explain.

(B) If you paired such pyramids randomly, and took the average height as a datum, what distribution would you expect among the data from all such pairings?

Answer 1

(A) Pyramids grow linearly more slowly as their volume increases. Therefore, the average pyramid is taller than the 27.5-year-reign pharoah's. (so technically, you are integrating t^(1/3)dt. The difference is about 16%).

(B) Individual data from ANY distribution, if paired randomly and averaged, result in a normal (Gaussian) distribution for the averaged data. This has been true since at least the time of the pharoahs. Coincidence??

-- Cheers, --Mr. d.