Quote:
Originally Posted by Ether
[b]OK I'm going to drop a bombshell here.
The correct answer to the OP, as worded, is 0.33634
...
Sampling segments by choosing the segment endpoint coordinates from a random uniform distribution does not match the probability density of the segment lengths in the infinite set.
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It's hard to disagree with your first figure.
If you can switch the order of integration and then integrate the pdf over theta, then you should have a pdf for L only. I'm trying to wrap my head around why that pdf for L should be different than the one effected by a uniform weighting of dx1dx2dy1dy2 differential elements. The concept in my head is that there's no invertible change of variables with which to set up that translation. The transformation from coordinates to length is not injective, so the Jacobian in the change of variables is zero: hard to divide by, conceivably giving the ratio between our answers in some sort of limit. Or from another view the four-dimensional differential element is of measure zero within the one-dimensional one, reflected by a pdf for the coordinates that is somehow undefined or not finite??
I'm very curious if there is an appropriate pdf for the coordinate approach that would give this result. Or is it just the wrong way to look at the problem?
This seems to boil down to the difference between:
[1] Average distance between all unordered pairs of points
[2] Average length of all line segments
But what's the difference?
While there are reasons to keep a float charge on your batteries, it may be like giving your son a flu shot when he has the measles. - rtfgnow [more] 
22-08-2016, 18:59
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