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:

• The Earth is a perfect sphere with radius 3,958.8 miles.
• 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?

Start with Brian's reorientation,  "We can always rotate the Earth so that one of our two random points is the "new" North Pole".  Then the average distance from the North pole to a random point on the earth must be the distance from the North Pole to the equator (6,215 miles), because for every point in the Northern Hemisphere there is a corresponding point in the Southern Hemisphere that has the same longitude and is equidistant from the equator, and the average of those two points along a great circle lies on the equator.  This approach does not even require us to assume that the earth is a perfect sphere, just that is is symmetrical with respect to any great circle. 

Edited on October 21, 2022, 6:38 pm

Posted by Steve Herman on 2022-10-21 18:04:22