Consider a circle with a circumference of n. On this circle, I define two arcs of length k &lt; n, A 1 and A 2 . The centres of the two arcs are uniformly distributed on the circle.Let Ω 1 = A 1 ∖ A 2 and Ω 2 = A 2 ∖ A 1 such that the length of Ω 1 ∪ Ω 2 is 2 k minus the overlapping part of the two arcs.What is the expectation of the length of Ω 1 ∪ Ω 2 ?

Question

Consider a circle with a circumference of n. On this circle, I define two arcs of length   k  &amp;lt;  n,       A    1   and       A    2  . The centres of the two arcs are uniformly distributed on the circle.Let       Ω          1        =      A    1    ∖      A    2   and       Ω          2        =      A    2    ∖      A    1   such that the length of       Ω    1    ∪      Ω    2   is   2  k minus the overlapping part of the two arcs.What is the expectation of the length of       Ω    1    ∪      Ω    2  ?

laure6237ma · Accepted Answer

By rotational symmetry, you can view the location of the first arc as fixed, and consider only the randomness of the second arc&#039;s position relative to the first one.If the [absolute] arc distance between the centers of the two arcs is   X, then you can check that the length of       Ω    1    ∪      Ω    2   is   2  min  {  X  ,  k  }. Then, note that   X is uniformly distributed between 0 and   n      /    2, so  E  [  2  min  {  X  ,  k  }  ]  =  2      ∫    0          n              /            2            1          n              /            2        min  {  t  ,  k  }    d  t  .I&#039;ll leave you to compute this integral, which will have different forms depending on whether   k  ≤  n      /    2 or   k  ≥  n      /    2.

Consider a circle with a circumference of n. On this circle, I define two arcs of length k <

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