Shooting randomly against a line in Euclidean and non-Euclidean spaces

In this paper we study a class of distributions related to the r.v. C-n (t) = ttan(1/n)Theta, for different distributions of Theta. The problem is related to the hitting point of a randomly oriented ray and generalizes the Cauchy distribution in different directions. We show that the distribution of C-n (t) solves the Laplace equation of order 2n, possesses even moments of order 2k < 2n - 1, and has bimodal structure when Theta is uniform. We study also a number of distributional properties of functionals of C-n (t), including those related to the arcsine law. Finally we study the same problem in the Poincare ' half- plane and this leads to the hyperbolic distribution Pr{h is an element of dw} = dw/pi cosh w of which the main properties are explored. In particular we study the distribution of hyperbolic functions of eta, the law of sums of i.i.d. r.v.'s eta(j) and the distribution of the area of random hyperbolic right triangles.