Laplace Type Problem with Non-uniform Distribution

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1 Applied Mathematical Sciences, Vol. 1, 16, no. 3, HIKARI Ltd, Laplace Type Problem with Non-uniform Distribution Giuseppe Caristi Department of Economics, University of Messina Via dei Verdi, 75, 981, Messina Italy Ersilia Saitta Department of Economics, University of Messina Via dei Verdi, 75, 981, Messina Italy Marius Stoka Sciences Academy of Turin Via Maria Vittoria, 3, 113, Torino, Italy Copyright c 16 Giuseppe Caristi et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Laplace type problems for different lattices have been considered intensively in the recent years. Particularly, in paper [6] the authors consider a Laplace type problem for a trapezoidal lattice with rectangle body test. In this paper we consider a lattice with fundamental cell composed by a trapezium as for the first time we consider as body test a random rectangle not uniformly distributed. We compute the probability that a random rectangle of constant sides intersects the a side of lattice when the position of the rectangle is a random variable with exponential and γ () distribution. Mathematics Subject Classification: 6D5, 5A Keywords: Geometric Probability, stochastic geometry, random sets, random convex sets and integral geometry

2 1596 Giuseppe Caristi et al. 1 Introducion In [1], [], [3], [4], [5] and [6] the authors consider several different Buffon- Laplace type problems. Starting from the results obtained by Caristi and Ferrara in [5] where for the first time the authors considered together with the traditional Buffon type problem also the different cases for the deformations of the considered lattice. Caristi and Molica Bisci [6] extended this method to a stochastic geometric problem on a circle. In [] Laplace type problems for a triangular lattice have been considered but the authors focused their attention considering different testing bodies on the same network and in the same time. In [1] the authors studied the problem of the different bodies test but considering different lattice with axial symmetry. Now, considering a fundamental cell composed by a trapezium we consider as body test a random rectangle not uniformly distributed. In fact we solve a Laplace type problem considering a random body test distributed according to an exponential distribution and in according to a γ() distribution. Main Results Let R (a, b; α) be a lattice with the fundamental cell C an trapezium with sides a < b and π 4 α π 3 By fig. 1 we have that: fig.1 areac = (b a ) tgα. (1) 4

3 Laplace type problem with non-uniform distribution 1597 Theorem 1 The probability that a random rectangle r of constant sides l, m with < m l < a intersects a side of the lattice R is: P int = (b a ) tgα f (ϕ) dϕ {(a + b) (l sin ϕ + m cos ϕ) + (b a) tgα (l cos ϕ + m sin ϕ) l [sin α sin ϕ + (1 + cos α) (1 cos ϕ)] sin α m sin ϕ + lm sin α [cos ϕ (1 + cos α) sin ϕ] + 3 } lm f (ϕ) dϕ, () where ϕ is the angle formed by the side of lenght l of the rectangle r with the line BC (or AD), the position of r is determined by its center and by the angle ϕ. Proof. We consider the limiting positions of r, for a specified value of ϕ, in the cell C. We obtain fig. and the formula By fig. we have: fig. 1 areaĉ (ϕ) = areac 4lm areaa i (ϕ). (3) i=1 areaa 1 (ϕ) = l sin ϕ sin (α ϕ), sin α

4 1598 Giuseppe Caristi et al. areaa (ϕ) = m ctg (α ϕ), areaa 3 (ϕ) = 1 [ ] b a [l sin (α ϕ) + m cos (α ϕ)] l sin ϕ + m cos ϕ cos α sin α 1 [ lm + m ctg (α ϕ) ], areaa 4 (ϕ) = m cos ϕ cos (α ϕ), sin α areaa 6 (ϕ) = 1 (l sin ϕ + m cos ϕ) [ b areaa 5 (ϕ) = l tgα, m cos (α ϕ) + l sin (ϕ + α) sin α areaa 7 (ϕ) = l sin ϕ sin (ϕ + α), sin α ] 1 ( l tgϕ + lm ), areaa 8 (ϕ) = m ctg (α + ϕ), areaa 9 (ϕ) = 1 ( ) b a [l sin (ϕ + α) m cos (ϕ + α)] l sin ϕ + m cos ϕ + cos α sin α m ctg (ϕ + α) lm, areaa 1 (ϕ) = m cos ϕ cos (ϕ + α), sin α areaa 11 (ϕ) = l tgϕ, areaa 1 (ϕ) = 1 (l sin ϕ + m cos ϕ) [ a Replacing these relations in (3) it follows that areaĉ (ϕ) = areac l sin ϕ m cos (ϕ + α) sin α ] l tgϕ lm. { 1 (a + b) (l sin ϕ + m cos ϕ) + (b a) tgα (l cos ϕ + m sin ϕ) l m [sin α sin ϕ + (1 + cos α) (1 cos ϕ)] sin ϕ+ (4) sin α lm 4 sin α [cos ϕ (1 + cos α) sin ϕ] lm}. Denoting by M the set of all the rectangles r which have their center in the cell C. We denote by N the set of the all rectangles r completely contained in C. In view of [8], we get:

5 Laplace type problem with non-uniform distribution 1599 P int = 1 µ (N) µ (M), (5) where µ is the Lebesgue measure in Euclidean plane. To compute the above measures we use the Poincaré kinematic measure [7]: dk = dx dy dϕ, where x, y are the coordinates of the center of r and ϕ the angle already defined. Considering that the direction of r is a random variable with density of probability f (ϕ), we have: µ (M) = f (ϕ) dϕ dxdy = {(x,y) C } (areac ) f (ϕ) dϕ = areac f (ϕ) dϕ, (6) and then µ (N) = f (ϕ) dϕ {(x,y) Ĉ(ϕ)} areac dxdy = f (ϕ) dϕ ( ) areaĉ (ϕ) f (ϕ) dϕ = { 1 (a + b) (l sin ϕ + m cos ϕ) + (b a) tgα (l cos ϕ + m sin ϕ) l m [sin α sin ϕ+ (1 + cos α) (1 cos ϕ)] sin ϕ+ sin α lm 4 sin α [cos ϕ (1 + cos α) sin ϕ] + 3 } 4 lm f (ϕ) dϕ, (7) P int = (b a ) tgα f (ϕ) dϕ {(a + b) (l sin ϕ + m cos ϕ) + (b a) tgα (l cos ϕ + m sin ϕ) l sin α [sin α sin ϕ + (1 + cos α) (1 cos ϕ)] m sin ϕ+ lm sin α [cos ϕ (1 + cos α) sin ϕ] + 3 } lm f (ϕ) dϕ. (8)

6 16 Giuseppe Caristi et al..1 Exponential random variable Considering f (ϕ) = e ϕ, by the change of variable e ϕ = u, we obtain In the same way, we have: f (ϕ) dϕ = 1 e α. (9) e ϕ sin ϕdϕ = 1 1 e α (sin α + cos α), e ϕ cos ϕdϕ = e α (sin α cos α), (1) and e ϕ sin ϕdϕ = 1 5 [ e α (sin α + cos α) ] e ϕ cos ϕdϕ = 1 5 [ 1 + e α ( sin α cos α) ]. (11) Replacing in (9) the relations (1) and (11) we obtain the following: Theorem The probability that a random rectangle r of constant sides l, m with < m l < a and distributed according to the exponential distribution, intersects a side of the lattice R is: P int = (b a ) tgα (1 e α ) { 1 [l (a + b) + m (b a) tgα] [ 1 e α (sin α + cos α) ] + 1 [m (a + b) + l (b a) tgα] [ 1 + e α (sin α cos α) ] [ e α (sin α + cos α) ] ( 1 + cos α sin α l + 1 sin α lm [ 1 + e α ( sin α cos α) ] + 3 lm ( 1 e α)}. ( l + m 1 + cos α ) ) sin α lm

7 Laplace type problem with non-uniform distribution 161. γ () random variable Considering now f (ϕ) = ϕe ϕ, we obtain 1 ϕe ϕ = 1 (1 + α) e α, ϕ sin ϕe ϕ dϕ = 1 1 e α cos α α e α (sin α + cos α), ϕe ϕ cos ϕdϕ = 1 e α (sin α + cos α) + α e α (sin α cos α) and ϕe ϕ sin ϕdϕ = e α (3 sin α + 8 cos α) + αe α (sin α cos α), ϕe ϕ cos ϕdϕ = e α (4 sin α + 9 cos α) + αe α (sin α cos α). We have: Theorem 3 The probability that a random rectangle r of constant sides l, m with < m l < a and distributed according to the γ () distribution, intersects a side of the lattice R is: P int = (b a ) tgα (1 e α ) { 1 [l (a + b) + m (b a) tgα] [ 1 e α cos α αe α (sin α + cos α) ] + 1 e α [m (a + b) + l (b a) tgα] [sin α + cos α + α (sin α cos α)] ( l + m cos α ) sin α lm [ ] 5 e α (3 sin α + 8 cos α) + αe α (sin α cos α) +

8 16 Giuseppe Caristi et al. ( 1 + cos α sin α l + 1 ) sin α lm [ ] 5 e α (4 sin α + 9 cos α) + αe α (sin α cos α) ( 1 + cos α sin α l 3 ) [1 lm ]} (1 + α) e α. References [1] D. Barilla, M. Bisaia, G. Caristi, A. Puglisi, On Laplace type problems (II), Far East Journal of Mathematical Sciences, 58 (11), no., [] D. Barilla, G. Caristi, A. Puglisi, M. Stoka, Laplace type problems for a triangular lattice and different body test, Applied Mathematical Sciences, 8 (14), no. 13, [3] U. Basel, A. Duma, A Laplace Type Problem for a Lattice of Rectangles with Triangular Obstacles, Applied Mathematical Sciences, 8 (14), no. 166, [4] V. Bonanzinga, L. Sorrenti, Geometric probabilities for cubic lattices with cubic obstacles, Suppl. Rend. Circ. Mat. Palermo, Serie II, 81 (9), [5] G. Caristi, M. Ferrara, On Buffon s problem for a lattice and its deformations, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 45 (4), no. 1, 13-. [6] G. Caristi, G. Molica Bisci, A problem of stochastic geometry on a circle, Far East Journal of Mathematical Sciences, 5 (7), no., [7] H. Poincaré, Calcul des Probabilités, ed., Gauthier Villars, Paris, 191. [8] M. Stoka, Probabilités géométriques de type Buffon dans le plan euclidien, Atti Acc. Sci. Torino, 11 ( ), Received: March 1, 16; Published: April 3, 16