91Ó°ÊÓ

A two-dimensional Poisson process is a process of randomly occurring events in the plane such that (i) for any region of area \(A\) the number of events in that region has a Poisson distribution with mean \(\lambda A\), and (ii) the number of events in nonoverlapping regions are independent. For such a process, consider an arbitrary point in the plane and let \(X\) denote its distance from its nearest event (where distance is measured in the usual Euclidean manner). Show that (a) \(P[X>t\\}=e^{-\lambda \pi t^{2}}\), (b) \(E[X]=\frac{1}{2 \sqrt{2}}\).

Consider a conditional Poisson process in which the rate \(L\) is, as in Example \(5.29\), gamma distributed with parameters \(m\) and \(p\). Find the conditional density function of \(L\) given that \(N(t)=n\).