= Solution
Write $s=e^{-\lambda t}$ for individual survival <probability>. Conditional on a clump of size $n$, its death <probability> is $(1-s)^n$. Averaging with the <geometric distribution> gives
$$
P(\text{dead})=(1-\theta)\sum_{n=1}^\infty\theta^{n-1}(1-s)^n
=\frac{(1-\theta)(1-s)}{1-\theta+\theta s}.
$$
Thus
$$
\boxed{p(t,\lambda)=\frac{s}{1-\theta+\theta s}.}
$$
Here $0\le\theta<1$ and $t>0$; different clumps and experimental repetitions are understood independent for the sampling model.
The live-clump counts have independent <binomial distributions>, all with the same $p$. Up to a constant, their combined <likelihood function> is $p^R(1-p)^{N-R}$, maximized at $\widehat p=R/N$ when $0<R<N$. Solving $p=s/(1-\theta+\theta s)$ for $s$ gives $s=p(1-\theta)/(1-\theta p)$. Therefore
$$
\boxed{\widehat\lambda=\frac1t\log\frac{N-\theta R}{R(1-\theta)}.}
$$
Monotonicity of $p$ in $\lambda$ makes this the <maximum-likelihood estimator>. At the sampling boundaries, $R=N$ gives $\widehat\lambda=0$, while $R=0$ gives only a supremum as $\lambda\to\infty$, with no finite maximizer. These are the limiting interpretations of the formula.
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