= Solution
For an interval of length $t$, the joint <probability generating function> of the marked counts is
$$
\mathbb E[s^{M^{(1)}}r^{M^{(2)}}]
=\exp\{\lambda t[ps+(1-p)r-1]\}
=\exp\{\lambda pt(s-1)\}\exp\{\lambda(1-p)t(r-1)\}.
$$
This 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 \b[the two processes are independent Poisson processes of rates $\lambda p$ and $\lambda(1-p)$]. For $m$ types the same calculation gives $\prod_j\exp[\lambda p_jt(s_j-1)]$, proving independent Poisson processes of rates $\lambda p_j$. A zero probability gives an identically zero process.
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