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

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 $1\le i\le n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of ${Z}_{i}$ that are at least $\frac{1}{n}$.

What is $E\left[X\right]$ and $Var\left[X\right]$?

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)
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Juliet Mcdonald
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.