Spatial Poisson Processes, and Relatives of Poisson Processes

Spatial Poisson Process

N = rpois(1, 0.1 * 20 * 10)
u1 = runif(N, 0, 20)
u2 = runif(N, 0, 10)
plot(u1, u2, pch = 20,
     xlab = "x-coordinate", ylab = "y-coordinate",
     main = "Spatial Poisson Process on 20x10 area with intensity 0.1")

Distance from center to closest point

n_rep = 10000
R = rep(NA, n_rep)

for (i in 1:n_rep) {
  N = rpois(1, 0.1 * 20 * 10)
  u1 = runif(N, 0, 20)
  u2 = runif(N, 0, 10)
  R[i] = min(sqrt((u1 - 10) ^ 2 + (u2 - 5) ^ 2))
}

hist(R, freq = FALSE)