Two points within the Arctic Circle are chosen at random, using a uniform distribution over the entire area. That is, any region of a given area is as likely as any other region with that area, to receive a given point.
What is the expected value of their great circle distance from each other?
Assume:
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The Earth is a perfect sphere with radius 3,958.8 miles.
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The Arctic Circle is located at 66.55° North.
Both calculus answers and simulation answers are welcome. As we are using approximations here, especially about the perfect sphericity of the Earth, the exactness of calculus is not really needed.
Part 2:
... and how about two points on the whole Earth?
Part 2 comment - after I read the word "easy" in BS's solution, I too realized that part 2 is just the average separation on a great circle. D'oh! Anyway, I had
simulated the two cases. The results are below. Since doing the whole globe, in addition to the cap, takes a lot longer, I include a final longer run with just the cap.
Num of pairs % pairs on cap ave miles globe ave miles cap
--------------------------------------------------------------------------------
100000 0.186 6218.922 1410.001
1000000 0.175 6215.264 1442.732
10000000 0.170 6218.219 1458.706
100000000 0.170 6218.347 1455.410
10000000 0.170 --- 1456.435
100000000 0.171 --- 1455.409
1000000000 0.171 --- 1457.298
So, globe: 6218.3 miles, cap: 1457.3 (+/-) 2 miles,
Or, better:
globe: 10,007.4 km, cap: 2345.3 km
Edited on October 22, 2022, 5:26 pm
soln
