Consider a collection of circles in the plane whose centers are distributed according to a spatial Poisson process with parameter $\lambda|A|$, where $|A|$ denotes the area of the set $A$. The radius of each circle is assumed to be a random variable independent of the location of the center of the circle, with density $f(r)$ and finite second moment.
(a) Show that $C(r)$, defined to be the number of circles that cover the origin and have centers at a distance less than $r$ from the origin, determines a variable-time Poisson process, where the time variable is now taken to be the distance $r$.
(b) Show that the number of circles that cover the origin is a Poisson random variable with parameter
$$\lambda \int_0^\infty \pi r^2 f(r)dr.$$
For part (a) there is a hint that says to prove that the probability of the event to occur between $r$ and $r+dr$ is
$$\lambda 2\pi rdr \int_r^\infty f(\rho)\,d\rho+o(dr).$$
I think I understand where the $2\pi\lambda r$ comes from, I believe that it is the probability of the event happening in $dr$ per unit area.
Then it says to show that $C(r)$ is a nonhomogeneous Poisson process with parameter
$$\lambda(r) = 2\pi\lambda r\int_r^\infty f(\rho)d\rho.\tag{$\ast$}$$I don't understand why $\int_r^\infty f(\rho)\,d\rho$ is there. I understand that it is that probability that the length of the radius is greater than or equal to $r$. But I don't understand why it is needed. I'm sure $o(dr)$ is some kind of error term.
For (b), I'm pretty you are supposed to make a "deterministic change" to turn this into a homogeneous Poisson process. Some solutions I have say that to get the parameter, integrate the function $(\ast)$,
$$\int_0^\infty\lambda(r)dr.\tag{$\star$}$$
My questions are:
- For the equation $(\ast)$, why do you need $\int_r^\infty f(\rho)\,d\rho$?
- For the integral $(\star)$, why are the limits of integration $0$ to $\infty$?
Best Answer
1.
$2\pi \lambda r dr + o(dr)$ is the probability of a circle of any radius occuring between the two circles of radii $r$ and $r+dr$. (I think the given hint expression is missing a $dr$.) The event required is that the circle covers the origin, which means it has radius at least $r$, hence the appearance of the factor $\int_r^\infty f(\rho)d\rho$.
2.
You're using the non-homogeneous Poisson process from part (a), say $X(r), r\geq 0$, and this has a Poisson distribution with parameter $\int_0^r \lambda(u)\;du$. That is,
$$P(X(r) = k) = \dfrac{\mu^k}{k!}e^{-\mu} \qquad\text{where } \mu=\int_0^r \lambda(u)\;du.$$
Ref: Poisson point process - non-homegeneous
The probability you want to check is $\lim_\limits{r\to\infty} P(X(r)=k)$, hence the integration limits $0$ to $\infty$.