My question is similar to this one but for rectangles instead of lines.
Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed random points inside the rectangle, and why?
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2$\begingroup$ Related: stats.stackexchange.com/q/22488/2970 $\endgroup$– cardinalOct 7, 2012 at 14:32
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1$\begingroup$ In the excellent related reference stats.stackexchange.com/q/22488/2970 @cardinal has not only calculated the average but also the distribution function of the distance. (+1) $\endgroup$– Dr. Wolfgang HintzeNov 2, 2022 at 22:08
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$\begingroup$ For the similar problem in a circle with radius a I obtain for the average distance L between the two random points the expression $\text{<L>} = \frac{128 a}{45 \pi}$, and for the average of the square of the distance I get $\text{<L^2>} = a^2$ $\endgroup$– Dr. Wolfgang HintzeNov 9, 2022 at 21:53
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$\begingroup$ For the similar problem in the volume of the unit sphere I found for the $k$-th moment of the distance the following simple expression $<D^k>=\frac{72\ 2^k}{(k+3) (k+4) (k+6)}$ $\endgroup$– Dr. Wolfgang HintzeNov 19, 2022 at 15:31
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$\begingroup$ For the similar problem in the volume of the unit sphere I found, using Fourier transform of the moment generating function, for the PDF of the distance D the surprisingly simple expression $PDF(D) = \frac{3}{16} (D+4) (2-D)^2 D^2$. The problem of the statistics of the distance of two random points in the volume of the unit sphere should thereby be completely solved. $\endgroup$– Dr. Wolfgang HintzeNov 20, 2022 at 14:51
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1 Answer
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The answer, given in (Burgstaller and Pillichshammer 2009; Mathai et al. 1999), is
$$ \frac1{15} \left( \frac{L_w^3}{L_h^2}+\frac{L_h^3}{L_w^2}+d \left( 3-\frac{L_w^2}{L_h^2}-\frac{L_h^2}{L_w^2} \right) +\frac52 \left( \frac{L_h^2}{L_w}\log\frac{L_w+d}{L_h}+\frac{L_w^2}{L_h}\log\frac{L_h+d}{L_w} \right) \right)\;, $$
where $d=\sqrt{L_w^2+L_h^2}$.
REFERENCES:
- Burgstaller, B. and Pillichshammer, F., "The average distance between two points", Bulletin of the Australian Mathematical Society, 80(3), pp.353-359, 2009.
- Mathai, A.M., Moschopoulos, P., Pederzoli, G., "Random points associated with Rectangles", Rendiconti del Circolo Matematico di Palermo, II, XLVIII (1999).
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1$\begingroup$ It seems to be a simple problem. But it's not. Good find. (+1) $\endgroup$– JACKY88Oct 7, 2012 at 14:18
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$\begingroup$ The famous book "Integral geometry and geometric probability" of Santalo is available on line at (projecteuclid.org/euclid.bams/1183539854). $\endgroup$ Oct 27, 2017 at 10:55