Question

It is the question on the text book but has no solution to it.

Suppose trees in a forest are distributed according to a Poisson process. Let X be the distance from an
arbitrary starting point to the nearest tree. The average number of trees per square metre is lambda". Derive
f(x) the same way we derived the Exponential probability density function. You are now using the
Poisson distribution in two dimensions (area) rather than one dimension (time).

I am wondering since lambda represents the average number of trees per square metre, how can we link this to the distance from an
arbitrary starting point to the nearest tree? In other word, how to solve this 2-dimension poisson question?

Thank you so much!

Answer 1

Consider the following equationThe exact solution of the previous equation is given asIn the view of the homotopy decomposition method, (11) can be first transformed toFollowing the decomposition techniques, we obtain the following equationComparing the terms of the same power of  leads toThe following solutions are obtained:In the same manner one can obtain the rest of the components. But for eight terms were computed and the asymptotic solution is given byTherefore in general for any  we have