Suppose N is a Poisson process on R2 with rate function λ(x, y) at the location (x, y). For instance, N could represent locations of certain types of animal nests, diseased trees, land mines, auto...

Suppose N is a Poisson process on R2 with rate function λ(x, y) at the location (x, y). For instance, N could represent locations of certain types of animal nests, diseased trees, land mines, auto accidents, houses of certain types of people, flaws on a surface, potholes, etc. Let Dn denote thedistance from the origin to the n-th nearest point of N. (a) Find an expression for the distribution and mean of D1. (b) Find an expression for the distribution of Dn when λ(x, y) = λ. (c) Are the differences Dn −Dn−1 independent (as they are for Poisson interpoint distances on R)? (d) Suppose there is a point located at (x∗, y∗). What is the distribution of the distance to the nearest point? (e) Is λ(x, y)=1/(x2 + y2)1/2 a valid rate function for N to be a Poisson process under our definition? (f) Specify a rate function λ(x, y) under which P{N(R2) <>