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)

Question

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 &#039;Probability and Computing&#039; by Mitzenmacher and Upfal)

Juliet Mcdonald · Accepted Answer

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.

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

Answered question

Answer & Explanation

New Questions in High school geometry