 Solution

Let $C$ be the clump size and put $q=e^{-\lambda t}$. The probability that all its members die, conditional on $C=n$, is $(1-q{)}^n$. Summing the geometric series givesSubtracting from one yields $\boxed{p(t,\lambda )=q/(1-\theta +\theta q)}$, for $0\le \theta <1$.

Assuming independent clumps and independent repeats, the combined likelihood is proportional to $p^R(1-p{)}^{N-R}$. For $0<R<N$ its unique maximizer is $\hat{p}=R/N$. The relation between $p$ and $q$ is one-to-one, with $q=(1-\theta )p/(1-\theta p)$. For $t>0$ this givesIf $R=N$, the estimate is the boundary value zero; if strictly positive rates are required, it is only a limiting supremum. If $R=0$, the likelihood approaches its supremum as $\lambda \to \infty$ and there is no finite estimate. These boundary cases explain when the printed logarithmic expression is finite.