Sunday, May 31, 2009

Amazing theorem

Dear Seminarians:

Yesterday I ran across the most amazing theorem. In WWII, Army Intelligence had the job of estimating the number of German Mark V tanks manufactured. The Germans naively numbered them sequentially, starting with "1" as they rolled off the assembly line. Problem: From a sample of M captured tanks, estimate the total number N that exist. This is possible to obtain with amazing precision!

Say you have a sample of captured tanks numbered {57, 103, 406, 44, 91}. That's a sample of five. So take 6/5 of the largest serial number and subtract 1. in this case, (6/5 X 406) -1 = 486 tanks total would be your best guess from doing this once.

The algorithm is to take the largest number out of a random sample of k tanks. Multiply this largest number by (k+1)/k and subtract 1. Now do it for another random set of k numbers. And again...and again...The mean of these numbers is an unbiased estimate which will converge to N, the total number! Assuming that the serial numbers of captured tanks was random to begin with, of course.

In real life, there were about 3000 tanks manufactured and Army Intelligence took random samples of size 100 out of the five hundred or so they captured many times. The mean of these numbers converged to 3001!


1 comment:

unsquander said...

This is the central limit theorem?