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.
By definition,
P(Y2>y |N(1)=10)
= 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]
You should probably use the fact that, conditionally toOriginally Posted by kingwinner
, the points of the Poisson process inside the circle of radius 1 have same joint distribution as 10 independent random variables uniformly distributed in the unit disk. (In particular,
depends only on these points) So you can forget about the Poisson process and only consider a situation involving uniform distribution in the unit disk.
hmm...I haven't encountered this theorem in my course, so I think I would have to do the computations by direct method.You should probably use the fact that, conditionally to
I know that N(y)~Poisson(λ pi y^2), and I know that for Poisson process, things happening in non-overlapping sets are independent. I think the use of these would allow me to compute directly.
But I am not sure how to express the event {Y2>y and N(1)=10} entirely in terms of N(y). Could you please give me some help on this?
[I don't really bother with all the computations, but I know, at least in theory, that if we can express the event {Y2>y and N(1)=10} entirely in terms of N(y), then the problem is basically solved.]
Thank you very much!
(in fact, this theorem is the exact analog to the theorem about ordered uniform random variables in the 1-dimensional case)
You could find the distribution of, you have:
=10)= P(N(r)\leq 1, N(1)=10)=)
and all these probabilities can be computed easily using annuli. For instance,is the probability that there is 1 point in
and 9 points in
, and since these sets are disjoint, the numbers of points therein are independent (and Poisson distributed with parameter proportional to the area...).
Finally, you can useto compute the expectation.
I let you try that.
Thanks!(in fact, this theorem is the exact analog to the theorem about ordered uniform random variables in the 1-dimensional case)
You could find the distribution of
and all these probabilities can be computed easily using annuli. For instance,
Finally, you can use
I let you try that.
If I use
E(Y2|N(1)=10) =
1
∫ y f(y) dy
0
where f(y) is the conditional density function of Y2 given that N(1)=10,
will I get the same answer?
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