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?
Asked
Active
Viewed 1.1k times
11
-
2Related: http://stats.stackexchange.com/q/22488/2970 – cardinal Oct 07 '12 at 14:32
-
1In 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) – Dr. Wolfgang Hintze Nov 02 '22 at 22:08
-
For the similar problem in a circle with radius a I obtain for the average distance L between the two random points the expression $\text{
} = \frac{128 a}{45 \pi}$, and for the average of the square of the distance I get $\text{<L^2>} = a^2$ – Dr. Wolfgang Hintze Nov 09 '22 at 21:53 -
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 $<D^k>=\frac{72\ 2^k}{(k+3) (k+4) (k+6)}$ – Dr. Wolfgang Hintze Nov 19 '22 at 15:31
-
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. – Dr. Wolfgang Hintze Nov 20 '22 at 14:51
-
I have finally found a reference covering the problem for the n-dimensional ball published in 2018: https://arxiv.org/pdf/math-ph/0004021.pdf. I'm happy to be able to confirm my findings. – Dr. Wolfgang Hintze Nov 28 '22 at 13:37
1 Answers
17
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).
-
1
-
The famous book "Integral geometry and geometric probability" of Santalo is available on line at (https://projecteuclid.org/euclid.bams/1183539854). – Jean Marie Oct 27 '17 at 10:55