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I have come across two forum about expected distance between two points and found seemingly conflicting results. Which one is correct? If they are two different ways of saying the same thing, can anyone help to explain?

These two discussions are:

  1. What is the expected distance between these two points? --> The result is L/4 (means 1/4 if the length is 1).

  2. Average Distance Between Random Points on a Line Segment --> The result is 1/3 because the discussion is focus on [0,1].

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    $\begingroup$ You've missed the assumption that in the first post, the second point is guaranteed to be to the left of the originally chosen point. There is no such assumption in the second post, and thus no conflict. (Please also read the tags you use in the future - algebraic-geometry was wholly inappropriate for this post, as you would have seen had you read the complete tag description.) $\endgroup$
    – KReiser
    Jan 19, 2020 at 21:08
  • $\begingroup$ What in this assumption makes the expected distance different? $\endgroup$
    – Amos Ku
    Jan 20, 2020 at 5:20
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    $\begingroup$ The fact it's to the left instead of potentially being on either side? I do not understand what you're getting at. Is there some reason you believe the answers should be the same? $\endgroup$
    – KReiser
    Jan 20, 2020 at 5:35

1 Answer 1

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The two questions are slightly different. In the first post, you take the second point to the left of the first one, and in the second post, you take the second point independently from the first one.

To see why it make a difference consider the three following cases and let's call the first point $x$ and the second point $y$:

  1. $x=0.5$. By symmetry, the expected distance to $y$ is the same whether or not $y$ restricted to the left of $x$
  2. $x=1$. $y$ is anyway to the left of $x$ so its expected distance to $x$ is the same in both cases
  3. $x=0$. Here, if $y$ is to the left of $x$, then it's distance to $x$ is zero. Otherwise it's expected distance to $x$ is 0.5.

Those three cases illustrate why in the first post the expected distance is smaller than in the second post.

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  • $\begingroup$ Intuitively, shouldn't the result be the same? It looks very weird to me to have two different expected distances when you viewpoint changes. $\endgroup$
    – Amos Ku
    Jan 19, 2020 at 23:42
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    $\begingroup$ Adding even a little bit of information to a problem regarding probability can change the results dramatically, so it shouldn't be too surprising that the answer changes when you add that the 2nd point must be to the left. This is pretty easy to check directly, too: if your first point is at $p\in[0,1]$ then the expected value of the distance to the 2nd point is $p/2$ if it's to the left and $p*(p/2)+(1-p)*(1-p)/2=p^2/2+(1-p)^2/2=p^2-p+1/2$ if it's location is unrestricted in $[0,1]$. Integrating over all $p$ gives the results. $\endgroup$
    – KReiser
    Jan 20, 2020 at 1:36
  • $\begingroup$ Please explained " if it's location is unrestricted in [0,1]. Integrating over all 𝑝 gives the results"... $\endgroup$
    – Amos Ku
    Jan 20, 2020 at 3:54
  • $\begingroup$ You can calculate the total expected distance as the average over all fixed first points $p$ of the expected distance to the second point. The way you average this quantity over all $p$ is to integrate. $\endgroup$
    – KReiser
    Jan 20, 2020 at 5:39
  • $\begingroup$ In many examples I have come across, expectation has the meaning of "the mean of data". What can we say about having two different "means" of the distance if we take different starting point? $\endgroup$
    – Amos Ku
    Jan 20, 2020 at 9:00

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