Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

Thanks!

share|improve this question
    
Also posted to MathOverflow mathoverflow.net/questions/85038/… –  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
add comment

3 Answers 3

up vote -1 down vote accepted

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).

share|improve this answer
add comment

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.)

share|improve this answer
    
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
add comment

Are you referring to the Parking Problem? See e.g. http://mathworld.wolfram.com/RenyisParkingConstants.html

share|improve this answer
    
Not quite, I'm thinking about DNA sequencing –  Tianyang Li Jan 4 '12 at 12:41
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.