CALCULATION OF EXPECTED DISTANCE ON A UNIT CUBE.

Transcription

1 CALCULATION OF EXPECTED DISTANCE ON A UNIT CUBE. JOHAN PHILIP Abstract. We calculate the expected distance between two random points on different faces of a unit cube. The problem is solved before, but we present an alternative method.. Introduction The expected distance between two random points on different sides of a square and the present analouge in three dimensions was treated numerically by James D. Klein, [Borwein et. al. 24, page 33]. Our attention to the problem was caught by [Bailey et. al. 25]. See also [Borwein et. al. 26] from which we cite problem 8: Calculate the expected distance between two random points on different faces of the unit cube. Hint: This can be expressed in terms of integrals as () E 3 := x2 + y 2 + (z w) 2 dw dx dy dz + (y u)2 + (z w) 2 du dw dy dz. Our method to find E 3 is to calculate the distribution functions for the two expressions under the square roots and then calculate the expectations. With some coaching, Maple is able to do the evaluations. Any higher moments can be calculated, when the distributions are known. The Maple worksheet cubedist.mw is available at 2. Notation and formulation. We start by calculating the distribution function F (s) = Prob(X 2 + Y 2 s) where X and Y are evenly distributed in a unit square and the distribution function G(t) = Prob((Z W ) 2 t) for Z and W evenly distributed in a unit square. Then, F and G are convolved to form the distribution H of the whole expression under the first square root. Convolving G by itself gives the distribution function K for the random part under the second square root. Key words and phrases. Distance on unit cube.

2 2 JOHAN PHILIP F(s) G(t) a sqrt(s) sqrt(t) b Figure. The areas F (s) and G(t). 3. The distributions F and G The probability that X 2 + Y 2 s equals the area in Fig. a. that is the intersection between the disk with radius = s and the unit square. We get πs, < s ; 4 (2) F (s) = s( 4 π arccos s ) + s, < s 2; 2 < s. The probability that (Z W ) 2 t equals the area of the diagonal strip in Fig. b. (3) G(t) = { ( t) 2, < t ; < t. 4. The first square root The probability that X 2 + Y 2 + (Z W ) 2 u is the convolution H of F and G H(u) = G(u s) df (s). Because of the various cases in the definitions of F and G, there are seven different integrals to calculate to get H. Maple does some of them directly and the remaining with the aid of the substitution s = u v 2. We get

3 DISTANCE ON UNIT CUBE 3 3 πu3/2 8 πu2, < u ; ( 3 + 5u u3/2 )π + ( 5u) u u2 arctan u, < u 2 ; 2 (4) H(u) = u + ( 3 + 5u u3/2 + 8 u2 )π + 5 (u + ) u 2 2u 3/2 arcsin ( ) 6 u 2 (u2 + u 3) arctan u 2, 2 < u 3;, 3 < u. The expectation of the first square root is E = 3 u dh(u). Told to use the substitution u = v 2, Maple evaluates this integral to E = 8 π ln 2 + ln ( + 2) (5) ln ( + 3) The second square root The probability that (Y U) 2 + (Z W ) 2 u is the convolution K of G by itself K(u) = G(u t) dg(t). Maple gives the following distribution, u, πu 8 3 u3/2 + 2 u2, < u ; (6) K(u) = 2u 3 2 u2 + 4(2u + ) u 3 2u arcsin ( u 2), < u 2 ; u, 2 < u. The expectation of the second square root is E 2 = 2 u + dk(u). Maple evaluates this integral directly to E 2 = π ln 2 + ln ( + 2) (7) ln ( + 3)

4 4 JOHAN PHILIP v~ Figure 2. The combined density function. 6. The combined distribution. Let V be the distance between two random points on different faces of a unit cube. We get the combined distribution C(v) = Probability( V v) as (8) C(v) = 4 5 H(v2 ) + 5 K(v2 ). The density function dc is shown in Figure 2. dv Combining E and E 2, we get the expection E(V ) = E 3 in () (9) E(V ) = 4 5 E + 5 E 2 = π ln ln ( + 2) ln ( + 3) Comment. We tested our method on the corresponding four-dimensional problem. When trying to calculate the four-dimensional H and K, we encountered integrals that we cannot solve like

5 DISTANCE ON UNIT CUBE 5 /a arcsin x ax + x x dx, where < a 2. References [] D.H. Bailey and J.M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, Vol. 53, Number 5, May 25 [2] D.H. Bailey, J.M. Borwein, V Kapoor, and E.W. Weisstein Ten Problems in Experimental Mathematics, dhbailey/dhbpapers/tenproblems.pdf, March 8, 26. [3] J.M. Borwein, D.H. Bailey, and R. Girgensohn Experimentation in Mathematics: Computational Paths to Discovery, A. K. Peters, Natick, MA, 24 Department of Mathematics, Royal Institute of Technology, S- 44 Stockholm Sweden address: johanph@kth.se