Let n points be placed uniformly at random on the boundary of a circle of circumference 1.

Micah Haynes 2022-05-09 Answered
Let n points be placed uniformly at random on the boundary of a circle of circumference 1.

These n points divide the circle into n arcs.

Let Z i for 1 i n be the length of these arcs in some arbitrary order, and let X be the number of Z i that are at least 1 n .

What is E [ X ] and V a r [ X ]?

Any hints will be appreciated. Thanks..

(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal)
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Answers (1)

Juliet Mcdonald
Answered 2022-05-10 Author has 16 answers
If you cut the circle along the first placed point, you can see that the situation is equivalent to taking the interval [0,1] and placing n 1 points uniformly at random into the interval.
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