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Tuesday, August 23, 2011

German Tanks and Drake 2-Bs: We Have the Number!

Hi Bill,

Finally getting back to you. I crunched the numbers...
I saw a total of 23 serial numbers reported. The important thing is that these numbers be reported somewhat randomly with no biases etc. I think this is the case, and the fact we have 23 numbers is very very good in terms of the power of this experiment. The highest number reported was 12955. Let m = 12955. The number of reports was 23 so let k = 23. The equation to use (from wikipedia) is below...
\hat{N} =m(1 + k^{-1}) - 1\,
Simply applying this equation we get an estimate highest serial number of 13517. So we can estimate that between 13000 and 14000 Drake 2Bs where made. I think I remember someone mentioning that the Drake 2B serial numbers did not start at zero. This is not really a problem. You just need to know at what number they did start, then subtract that number from 13517, and that would be the number of Drake 2Bs ever manufactured.

The lowest serial number reported to SolderSmoke was 2008, so you wouldn't need to subtract more than that.
You can think of this equation intuitively (a very SolderSmoke thing to do!). Imagine what happens when we have a single observation. k = 1, so our estimate is about 2 times what our highest observation is. This makes sense because you would guess your observation is most likely to be about half way between 0 and the true top number. If k = 2, then our estimate is about 1.5 times our highest observation. If k = 3, then our estimate is about 1.333 times our highest observation.... as we observe more numbers, we are more likely to have observed the top number so as k goes to infinity, our estimate moves towards our top observed number, which it should.

I hope my explanation made sense. Anyway I highly recommend SolderSmoke listeners who want to know more, to read the wikipedia page. Its quite well written and offers a lot for people who like hard formal explanations and an intuitive description. Keep up the good work!

I love the podcast - hopefully we may catch each other one day on the bands.

Scott (K6AUS)

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