# Alternate characterization of a spatial Poisson point process

Would I be correct to assess that a spatial Poisson point process on some compact, say the $$d$$-dimensional sphere, can be simulated by first choosing some $$n \sim \mathrm{Poisson} (\beta)$$ number of points, and then uniformly distributing them over this compact? Here $$\beta$$ is the mean number of points generated by the PPP.

Or in other words, conditioning the total number of points to be fixed as $$k$$, does a spatial Poisson point process simulate the uniform distribution with $$k$$-points over the compact?