Kavli Affiliate: George Clark
| First 5 Authors: George Clark, Alex Gonye, Steven J Miller, ,
| Summary:
During World War II the German army used tanks to devastating advantage. The
Allies needed accurate estimates of their tank production and deployment. They
used two approaches to find these values: spies, and statistics. This note
describes the statistical approach. Assuming the tanks are labeled
consecutively starting at 1, if we observe $k$ serial numbers from an unknown
number $N$ of tanks, with the maximum observed value $m$, then the best
estimate for $N$ is $m(1 + 1/k) – 1$. This is now known as the German Tank
Problem, and is a terrific example of the applicability of mathematics and
statistics in the real world. The first part of the paper reproduces known
results, specifically deriving this estimate and comparing its effectiveness to
that of the spies. The second part presents a result we have not found in print
elsewhere, the generalization to the case where the smallest value is not
necessarily 1. We emphasize in detail why we are able to obtain such clean,
closed-form expressions for the estimates, and conclude with an appendix
highlighting how to use this problem to teach regression and how statistics can
help us find functional relationships.
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