Laplace type problem with non-uniform distribution

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.