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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?

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