Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 26J / e / Solution

Assign each arriving car independently to a uniformly chosen lot. The Poisson thinning theorem gives three independent Poisson processes, each of rate $\lambda /3$. For an interval of length $t$, their joint probability generating function isThe factorization proves independence of the counts in that interval; independence over disjoint intervals proves independence of the processes.