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Continuing this question:

I sample two points from the $n$-dimensional unit cube: $$p_{i,1}, p_{i,2} \sim U([0, 1]^n)$$

Now I do this $N$ times. I define the maximum distance as

$$m_d := \max_{i=1,..., N}(|p_{i,1} - p_{i,2}|_2)$$

What is the expected maximum distance $\mathbb{E}(m_d)$ for $N$ pairs of 2 points?

Martin Thoma
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    What justifies your swapping expectation and square root in the second equality? – Clement C. Feb 10 '18 at 20:25
  • @ClementC. Ooh... damn... ok, then I really don't know how to solve this – Martin Thoma Feb 10 '18 at 20:32
  • Related: https://math.stackexchange.com/questions/1976842/how-is-the-distance-of-two-random-points-in-a-unit-hypercube-distributed?noredirect=1&lq=1 https://math.stackexchange.com/questions/1254129/average-distance-between-two-random-points-in-a-square/1254154#1254154 https://martin-thoma.com/average-distance-of-points/ (funny. Just realized one question was by you, and this is actually a bog post of yours.) – Clement C. Feb 10 '18 at 20:40
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    Is this your problem? I'm a bit confused by your usage of maximum distance. – orlp Feb 10 '18 at 21:00
  • Intuitively, the mean is $1/3$ for $n=1$ (two randomly chosen points in $(0,1).$) On average the smaller is at $1/3$ and the larger is at $2/3.$ And the proof is easy. The variance of the distance is a little harder to get because the min and max are not independent. – BruceET Feb 10 '18 at 21:03
  • @orlp With maximum distance, I mean the following: Suppose I sample 100 times $x_{i,1}$ and $x_{i,2}$. What is the expected maximum distance $d = \max_{i=1,..., 100}(|x_{i,1} - x_{i,2}|_2)$ I see? - I've Updated the question. – Martin Thoma Feb 10 '18 at 22:51
  • @ClementC. Hahaha, I couldn't find that question anymore. But looking at it, it seems not to be a duplicate. Just an extension. – Martin Thoma Feb 10 '18 at 23:02
  • @ClementC. And yes, the blog-post is super related. I want to extend it^^ – Martin Thoma Feb 10 '18 at 23:05

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