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There has been questions/videos that calculate the "distance between two random points" on a square/circle etc.

Every instance that I have seen so far all is done is to average (arithmetic mean) of the distances between all the points (by using integrals). My question is there an instance where the average of all distances and distance of two random points are not the same value? or the titles are misleading and instead of a title like "Random distance between two points on a square/circle" it should really be "Average distance between two points on a square/circle"?

jimjim
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    Just for completeness, can you link to some examples of the "random distance" terminology being used? – Troposphere Aug 21 '21 at 01:21
  • @Troposphere : just googling "random distance on circle/square" gives number of video/web pages, e.g. https://math.stackexchange.com/questions/1294800/average-distance-between-two-randomly-chosen-points-in-unit-square-without-calc is that what you mean? or this one : https://www.youtube.com/watch?v=i4VqXRRXi68 – jimjim Aug 21 '21 at 01:27
  • @Troposphere : Ahaa, yes, "Random distance" was misleading and meaningless, thank you for the hint to fix it up. – jimjim Aug 21 '21 at 01:31
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    It's important to remember that your notion of randomness ultimately depends on the pdf governing the observed points that are being sampled. For example, if $P,Q\sim \mathcal{U}( {\mathbb{D}})$ are iid where $\mathbb{D}$ is the unit disc, then the expected value of $|P-Q|$ is going to be different than if you sampled points $P,Q \sim f$ where $f(x,y)=\frac{2}{\pi}(x^2+y^2)\cdot 1_{\mathbb{D}}$ which gives preference to points closer to the unit circle. – Matthew H. Aug 21 '21 at 01:37
  • @MatthewPilling : what does pdf means ? – jimjim Aug 31 '21 at 02:29
  • @jimjim pdf means probability denisty function – Matthew H. Aug 31 '21 at 02:34

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The distribution function of the distance between two points $X$ and $Y$, inside some probability space $\Omega $, is given by $F(c):=\Pr [d(X,Y)\leqslant c]$, for some distance function $d$ in $\Omega $. By example, if $X$ and $Y$ are points chosen randomly in the unit radius disk in the plane, then usually $X$ and $Y$ are assumed to have an uniform distribution and to be independent, and the usual distance function is the Euclidean norm. Then if $D:=\{(x,y)\in \mathbb{R}^2: x^2+y^2\leqslant 1\}$ is the unit disk then the desnities of $X$ and $Y$ are $f_X=f_Y=\pi^{-1}\mathbf{1}_{D}$, and so $f_{X,Y}(s,t)=\pi^{-2}\mathbf{1}_{D}(s)\mathbf{1}_{D}(t)$, therefore

$$ F(c)=\int_{R_c}f_{X,Y}(s,t)\mathop{}\!d (s,t),\quad \text{ where }R_c:=\{(s,t)\in \mathbb{R}^2\times \mathbb{R}^2:\|s-t\|_2\leqslant c\} $$

I dont know if this clarify your doubt.

Masacroso
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  • It gives me a starting point to try to understand when they might coincide or not, and what are the conditions for the two to have the same value. – jimjim Aug 21 '21 at 02:00