Applied Mathematical Sciences, Vol. 10, 2016, no. 32, 1595 - 1602
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2016.6260
Laplace Type Problem with
Non-uniform Distribution
Giuseppe Caristi
Department of Economics, University of Messina
Via dei Verdi, 75, 98122, Messina Italy
Ersilia Saitta
Department of Economics, University of Messina
Via dei Verdi, 75, 98122, Messina Italy
Marius Stoka
Sciences Academy of Turin
Via Maria Vittoria, 3, 10123, Torino, Italy
c 2016 Giuseppe Caristi et al. This article is distributed under the Creative
Copyright 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 (2) distribution.
Mathematics Subject Classification: 60D05, 52A22
Keywords: Geometric Probability, stochastic geometry, random sets, random convex sets and integral geometry
1596
1
Giuseppe Caristi et al.
Introducion
In [1], [2], [3], [4], [5] and [6] the authors consider several different BuffonLaplace 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 [2] 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(2) distribution.
2
Main Results
Let < (a, b; α) be a lattice with the fundamental cell C0 an trapezium with
sides a < b and π4 ≤ α ≤ π3
fig.1
By fig. 1 we have that:
areaC0 =
(b2 − a2 ) tgα
.
4
(1)
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Laplace type problem with non-uniform distribution
Theorem 1 The probability that a random rectangle r of constant sides l, m
with 0 < m ≤ l < a2 intersects a side of the lattice < is:
Pint =
Z
2
(b2
−
a2 ) tgα
Rα
0
f (ϕ) dϕ
·
α
{(a + b) (l sin ϕ + m cos ϕ) + 2 (b − a) tgα (l cos ϕ + m sin ϕ) −
0
l2
[sin α sin 2ϕ + (1 + cos α) (1 − cos 2ϕ)] −
sin α
lm
3
2
m sin 2ϕ +
[cos 2ϕ − (1 + cos α) sin 2ϕ] + lm f (ϕ) dϕ,
2 sin α
2
(2)
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 C0 . We obtain fig. 2
fig. 2
and the formula
b0 (ϕ) = areaC0 − 4lm −
areaC
12
X
areaai (ϕ) .
i=1
By fig. 2 we have:
areaa1 (ϕ) =
l2 sin ϕ sin (α − ϕ)
,
2 sin α
(3)
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Giuseppe Caristi et al.
m2
ctg (α − ϕ) ,
2
1
b − a l sin ϕ + m cos ϕ
areaa3 (ϕ) = [l sin (α − ϕ) + m cos (α − ϕ)]
−
−
2
cos α
sin α
1
lm + m2 ctg (α − ϕ) ,
2
m2 cos ϕ cos (α − ϕ)
areaa4 (ϕ) =
,
2 sin α
l2
areaa5 (ϕ) = tgα,
2
1
1
m cos (α − ϕ) + l sin (ϕ + α)
areaa6 (ϕ) = (l sin ϕ + m cos ϕ) b −
− l2 tgϕ + lm ,
2
sin α
2
areaa2 (ϕ) =
l2 sin ϕ sin (ϕ + α)
,
2 sin α
m2
areaa8 (ϕ) = − ctg (α + ϕ) ,
2
b−a
l sin ϕ + m cos ϕ
1
−
areaa9 (ϕ) = [l sin (ϕ + α) − m cos (ϕ + α)]
+
2
2 cos α
sin α
areaa7 (ϕ) =
lm
m2
ctg (ϕ + α) −
,
2
2
m2 cos ϕ cos (ϕ + α)
areaa10 (ϕ) = −
,
2 sin α
l2
areaa11 (ϕ) = tgϕ,
2
2
1
l sin ϕ − m cos (ϕ + α)
l
lm
areaa12 (ϕ) = (l sin ϕ + m cos ϕ) a −
− tgϕ − .
2
sin α
2
2
Replacing these relations in (3) it follows that
b0 (ϕ) = areaC0 −
areaC
1
(a + b) (l sin ϕ + m cos ϕ) + (b − a) tgα (l cos ϕ + m sin ϕ) −
2
l2
m2
[sin α sin 2ϕ + (1 + cos α) (1 − cos 2ϕ)] −
sin 2ϕ+
(4)
2 sin α
2
lm
3
[cos
2ϕ
−
(1
+
cos
α)
sin
2ϕ]
+
lm
.
4 sin α
4
Denoting by M the set of all the rectangles r which have their center in the
cell C0 . We denote by N the set of the all rectangles r completely contained
in C0 . In view of [8], we get:
1599
Laplace type problem with non-uniform distribution
µ (N )
,
(5)
µ (M )
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,
Pint = 1 −
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:
Zα
ZZ
µ (M ) = f (ϕ) dϕ
dxdy =
0
{(x,y)∈C0 }
Zα
Zα
(areaC0 ) f (ϕ) dϕ = areaC0
0
f (ϕ) dϕ,
(6)
0
and
Zα
µ (N ) =
ZZ
f (ϕ) dϕ
0
dxdy =
Zα b0 (ϕ) f (ϕ) dϕ =
areaC
0
{(x,y)∈Cb0 (ϕ)}
Zα
areaC0
f (ϕ) dϕ−
0
α
Z
0
1
(a + b) (l sin ϕ + m cos ϕ) + (b − a) tgα (l cos ϕ + m sin ϕ) −
2
l2
m2
[sin α sin 2ϕ+ (1 + cos α) (1 − cos 2ϕ)] −
sin 2ϕ+
2 sin α
2
lm
3
[cos 2ϕ − (1 + cos α) sin 2ϕ] + lm f (ϕ) dϕ,
4 sin α
4
then
Pint =
Z
(7)
2
Rα
·
(b2 − a2 ) tgα 0 f (ϕ) dϕ
α
{(a + b) (l sin ϕ + m cos ϕ) + 2 (b − a) tgα (l cos ϕ + m sin ϕ) −
0
l2
[sin α sin 2ϕ + (1 + cos α) (1 − cos 2ϕ)] − m2 sin 2ϕ+
sin α
lm
3
[cos 2ϕ − (1 + cos α) sin 2ϕ] + lm f (ϕ) dϕ.
2 sin α
2
(8)
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2.1
Giuseppe Caristi et al.
Exponential random variable
Considering
f (ϕ) = e−ϕ ,
by the change of variable e−ϕ = u, we obtain
Zα
f (ϕ) dϕ = 1 − e−α .
(9)
0
In the same way, we have:
Zα
e−ϕ sin ϕdϕ =
1 1 −α
− e (sin α + cos α) ,
2 2
e−ϕ cos ϕdϕ =
1 1 −α
+ e (sin α − cos α) ,
2 2
0
Zα
(10)
0
and
Zα
e−ϕ sin 2ϕdϕ =
1
2 − e−α (sin 2α + 2 cos 2α)
5
e−ϕ cos 2ϕdϕ =
1
1 + e−α (2 sin 2α − cos 2α) .
5
0
Zα
(11)
0
Replacing in (9) the relations (10) and (11) we obtain the following:
Theorem 2 The probability that a random rectangle r of constant sides l, m
with 0 < m ≤ l < a2 and distributed according to the exponential distribution,
intersects a side of the lattice < is:
Pint =
2
(b2
−
a2 ) tgα (1
− e−α )
1
[l (a + b) + 2m (b − a) tgα] 1 − e−α (sin α + cos α) +
2
1 2
1
1 + cos α
−α
2
[m (a + b) + 2l (b − a) tgα] 1 + e (sin α − cos α) +
l +m −
lm
2
5
2 sin α
1 1 + cos α 2
1
−α
2 − e (sin 2α + 2 cos 2α) +
l +
lm
5
sin α
2 sin α
3
−α
−α
1 + e (2 sin 2α − cos 2α) + lm 1 − e
.
2
Laplace type problem with non-uniform distribution
2.2
1601
γ (2) random variable
Considering now
f (ϕ) = ϕe−ϕ ,
we obtain
Zα
ϕe−ϕ = 1 − (1 + α) e−α ,
1
Zα
ϕ sin ϕe−ϕ dϕ =
α
1 1 −α
− e cos α − e−α (sin α + cos α) ,
2 2
2
0
Zα
1
α
ϕe−ϕ cos ϕdϕ = e−α (sin α + cos α) + e−α (sin α − cos α)
2
2
0
and
Zα
ϕe−ϕ sin 2ϕdϕ =
4
1
+ e−α (3 sin 2α + 8 cos 2α) + αe−α (sin 2α − 2 cos 2α) ,
5 25
0
Zα
ϕe−ϕ cos 2ϕdϕ =
1
1
+ e−α (4 sin 2α + 9 cos 2α) + αe−α (sin 2α − cos 2α) .
5 25
0
We have:
Theorem 3 The probability that a random rectangle r of constant sides l, m
with 0 < m ≤ l < a2 and distributed according to the γ (2) distribution, intersects a side of the lattice < is:
Pint =
2
(b2 − a2 ) tgα (1 − e−α )
1
[l (a + b) + 2m (b − a) tgα] 1 − e−α cos α − αe−α (sin α + cos α) +
2
1 −α
e [m (a + b) + 2l (b − a) tgα] [sin α + cos α + α (sin α − cos α)] −
2
1 + cos α
2
2
l +m +
lm
2 sin α
4
1 −α
−α
+ e (3 sin 2α + 8 cos 2α) + αe (sin 2α − 2 cos 2α) +
5 25
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Giuseppe Caristi et al.
1
1 + cos α 2
l +
lm
sin α
2 sin α
1
1 −α
−α
+ e (4 sin 2α + 9 cos 2α) + αe (sin 2α − cos 2α) −
5 25
1 + cos α 2 3
l − lm 1 − (1 + α) e−α .
sin α
2
References
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(II), Far East Journal of Mathematical Sciences, 58 (2011), no. 2, 145-155.
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triangular lattice and different body test, Applied Mathematical Sciences,
8 (2014), no. 103, 5123-5131. http://dx.doi.org/10.12988/ams.2014.46423
[3] U. Basel, A. Duma, A Laplace Type Problem for a Lattice of Rectangles
with Triangular Obstacles, Applied Mathematical Sciences, 8 (2014), no.
166, 8309-8315. http://dx.doi.org/10.12988/ams.2014.411918
[4] V. Bonanzinga, L. Sorrenti, Geometric probabilities for cubic lattices with
cubic obstacles, Suppl. Rend. Circ. Mat. Palermo, Serie II, 81 (2009), 4753.
[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 (2004), no. 1, 13-20.
[6] G. Caristi, G. Molica Bisci, A problem of stochastic geometry on a circle,
Far East Journal of Mathematical Sciences, 25 (2007), no. 2, 367-374.
[7] H. Poincaré, Calcul des Probabilités, ed. 2, Gauthier Villars, Paris, 1912.
[8] M. Stoka, Probabilités géométriques de type Buffon dans le plan euclidien,
Atti Acc. Sci. Torino, 110 (1975-1976), 53-59.
Received: March 1, 2016; Published: April 30, 2016