Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 26J / a

For an interval of length $t$, the joint probability generating function of the marked counts isThis factors into two Poisson generating functions. Independent parent increments on disjoint intervals and independent marks give independent increment vectors, so the factorization holds jointly over all intervals. Thus the two processes are independent Poisson processes of rates $\lambda p$ and $\lambda (1-p)$. For $m$ types the same calculation gives ${\prod }_j\exp [\lambda p_{j}t(s_j-1)]$, proving independent Poisson processes of rates $\lambda p_j$. A zero probability gives an identically zero process.