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Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).

But the problem when the circle is changed to a line segment doesn't seem to have been studied before.

I'd like to know if there's any work out there who already obtained the probability distribution of the number and the length of the connected line segments that you get when randomly covering a line segment with another set of shorter segments, which may all be of equal length or have some kind of distribution.


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Also posted to MathOverflow… – user16299 Feb 28 '12 at 6:23
In future, if you are going to post to both sites, PLEASE indicate this in your question. That way we avoid duplicated effort. – user16299 Feb 28 '12 at 6:23

Are you referring to the Parking Problem? See e.g.

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Not quite, I'm thinking about DNA sequencing – Tianyang Li Jan 4 '12 at 12:41
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This problem can actually be solved using the exact same method as Chapter 4 of Solomon's geometric probability by using the inclusion-exclusion principle in a similar fashion. A brief outline is available here (although it may contain small errors).

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Invariance principle for the coverage rate of genomic physical mappings might interest you, if only for its list of references. (Caveat: I am the author.)

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I'm actually thinking more about results more exact than the Lander-Waterman model (when the sequence being sequenced is shorter) – Tianyang Li Jan 10 '12 at 5:41

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