Simulating a Boolean model in a uniform square

Question

I am looking to simulate a Boolean model with specified Poisson point process.

λ = 50;
pts = RandomPoint[
        Rectangle[], 
        RandomVariate[PoissonDistribution[λ*Integrate[1, {x, y} ∈ R]]]
      ];
Show[RegionPlot[Rectangle[]], Graphics[Point[pts]]]

Now I am trying to attach the grains using properties of an independently marked point process, so my attempt at a new code is now:

first attempt:

λ = 50; n = 50;
pts = RandomPoint[Rectangle[], 
   RandomVariate[
    PoissonDistribution[λ*Integrate[1, {x, y} ∈ R]]*
     UniformDistribution[n*{x, y} ∈ R]]];
Show[RegionPlot[Rectangle[]], Graphics[Point[pts]]]

second attempt:

λ = 50; r = UniformDistribution[];
pts = RandomPoint[
        Rectangle[], 
        RandomVariate[PoissonDistribution[λ*Integrate[1, {x, y} ∈ R]]]
      ];
Show[RegionPlot[Rectangle[]], Graphics[Circle[pts,r]]]

Neither of which work. I was hoping to be able to nest already built in functions, as my pure function writing skills aren't that good. My idea was that the independently marked p.p. is the product of the spacial p.p. times the distribution of the associated marks:

$\tilde{Φ}_{x_i,Ξ_i} = Φ_{x_i}*F(Ξ_i)$

Where the associated marks are the grains of the BM that are random closed circles $Ξ(m)=B_{x_i}(m)$ with random radius m, centered at the each point $x_i$.

Answer 1

I post this for fun but perhaps it may be useful.

vm = VoronoiMesh[RandomReal[10, {20, 2}], 
   MeshCellStyle -> {{1, All} -> Black, {2, All} -> LightYellow}];
pg = DiscretizeRegion /@ MeshPrimitives[vm, 2];
f[p_, r_] := 
  With[{v = RandomVariate[PoissonDistribution[r Area[p]]]}, 
   If[v == 0, {}, {Blue, Point[RandomPoint[p, v]], Red, 
     Text[Style[v, Bold], RegionCentroid[p]]}]];
Manipulate[g = Graphics[f[#, r] & /@ pg];
 Show[vm, g], {r, 0.1, 10}]