I have a Poisson point process (PPP) $\Phi$ with intensity $\lambda_M$ (density in Euclidean plane). If $R_m$ denotes the distance from a reference point, then what does the following equation means? As per my understanding, $r$ is some random variable such that $0\leq r \leq R_m$.
PDF of $R_m$ is given (in the research paper I am reading) by:
(btw, I did not find this formula on the web, and the research article has not shared any reference either. Maybe it is too well known, but not for me)
$$f_{R_m}(r) = 2\pi \lambda_M\ r\ \exp(-\lambda_M\pi\ r^2)$$
Since $\lambda_m$ is the density in the Euclidean plane, should the PDF formula be:
$$f_{R_m}(r) = (\lambda_M\ \pi r^2)^n \exp(-\lambda_M\pi\ r^2)/n!$$