Consider a Poisson process of rate λ on R^2. Let N(r) be the number of points in a circle of radius r centered at the origin and Y2 be the distance from the origin to the 2nd closest point.
Calculate E(Y2) and E[Y2 |N(1)=10].
I have no problem with computing E(Y2), but I am stuck with calcuating E[Y2 |N(1)=10].
So my first step is to try to compute the tail distribution function P(Y2>y |N(1)=10), and then differentiate to get the density function and then take the conditional expectation using the density.
= P(Y2>y and N(1)=10) / P(N(1)=10)
Now I am having trouble computing P(Y2>y and N(1)=10). How can we express it in terms of N(y) (which is the number of points in a circle of radius y about the origin)?
Can someone please go through the idea of how to compute this?
Any help is greatly appreciated!
[note: also under discussion in talk stats forum]