6.124 Suppose that plants of a certain species are randomly dispersed over a region, with a mean density of plants per unit area. That is, the number of plants in a region of area A has a Poisson distribution with mean . For a randomly selected plant in this region, let R denote the distance to the nearest neighboring plant.

a Find the probability density function for R.

[Hint: Note that P(R  r) is the same as the probability of seeing no plants in a circle of radius r.]

b Find E(R).