Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 28K (Applied Probability)

Let $N_j(A)$ be the number of particles of gas $j$ in a bounded Lebesgue measurable set $A$. Because Lebesgue measure is diffuse, two independent spatial Poisson point processes have no common particle almost surely. Hence the mixture count isWriting ${\lambda }_j=z_j|A|$, independence gives the probability generating functionThus

If $A_1,\dots ,A_m$ are pairwise disjoint, the familyis independent: counts in disjoint sets are independent within each gas, and the two gases are independent. Therefore the sums $N(A_r)$ are independent. These two facts prove directly that the mixture is a Poisson point process. By the Superposition theorem for Poisson point processes, its activity is

Put ${\lambda }_i=z|V_i|$. Thenand ${\lambda }_i\to \infty$. The Poisson central limit theorem givesSince $\sqrt{{\lambda }_i}=\sqrt{z}\sqrt{|V_i|}$,

Equivalently, the characteristic function of the left-hand side iswhich is the characteristic function of that centered normal distribution.

LetThe compact support of $g$ ensures that this sum has only finitely many nonzero terms almost surely. The intensity measure of the ideal gas isApplying the first-moment identity in Campbell theorem gives

For the second moment, separate equal and distinct particles:Subtracting $(\mathbb{E}{S}_g{)}^2$ yieldsThese are the mean and variance formulas in Campbell's theorem for a homogeneous Poisson point process.