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?
(In reply to
re: soln by Steve Herman)
Thanks Steve,
That's a cool fact about the Earth circumference equaling almost exactly 40,000 km. I did't know it, which, being an astronomer, is a bit embarrassing. (But hey, I do extragalactic....)
Using the average Earth radius as given by the problem, a great circle comes to 40030.6 km, and a perfect :-) simulation therefore should get a mean distance of 10,006.39 km. The simulation I ran with 10^8 points got 10,007.46 km, which I think is pretty good.
BTW - I tightened up the code a little to consider -nothing- below the arctic circle (previously, 99.8% of the pairs failed to qualify, and now all qualify) and I got these newest polar cap results:
points dist miles dist km
----------------------------------
10^7 1457.474 2345.577
10^8 1457.603 2345.784
10^9 1457.585 2345.756
So, things appear to have converged pretty well.
I thought about the integrals needed to get the exact value and
decided not only would this require W-alpha's help, but even then the quadruple integrals could still cause an overload. And, i skipped it.
Edited on October 22, 2022, 5:30 pm
re(2): soln
