Suppose that points are distributed over the half-line [0;1) according to a Poisson process of rate _. A sequence of independent and identically distributed nonnegative random variables Y1;Y2; : : : is used to reposition the points so that a point formerly at location Wk is moved to the location Wk CYk. Completely describe the distribution of the relocated points.  Suppose that particles are distributed on the surface of a circular region according to a spatial Poisson process of intensity _ particles per unit area. The polar coordinates of each point are determined, and each angular coordinate is shifted by a random amount, with the amounts shifted for distinct points being independent random variables following fixed probability distribution. Show that at the end of the point movement process, the points are still Poisson distributed over the region.

Suppose that points are distributed over the half-line [0;1) according to a Poisson process of rate _. A sequence of independent and identically distributed nonnegative random variables Y1;Y2; : : : is used to reposition the points so that a point formerly at location Wk is moved to the location Wk CYk. Completely describe the distribution of the relocated points.

 Suppose that particles are distributed on the surface of a circular region according to a spatial Poisson process of intensity _ particles per unit area. The polar coordinates of each point are determined, and each angular coordinate is shifted by a random amount, with the amounts shifted for distinct points being independent random variables following fixed probability distribution. Show that at the end of the point movement process, the points are still Poisson distributed over the region.