= Solution
For the fixed observation interval, $N_3$ has <Poisson distribution> with mean and <variance> $3\lambda$. Hence $N_3/3$ is unbiased and has <variance> $\lambda/3$. Its <score function> is $N_3/\lambda-3$, and its <Fisher information> is $3/\lambda$. The <Cramér-Rao bound> for unbiased estimation of $\lambda$ therefore gives <variance> at least $\lambda/3$, attained by $N_3/3$.
For the stopping-time observation, $T_{10}$ has <Gamma distribution> with shape $10$ and rate $\lambda$. Direct integration of its density gives
$$
\mathbb E_\lambda(T_{10}^{-1})=\frac\lambda9,\qquad
\mathbb E_\lambda(T_{10}^{-2})=\frac{\lambda^2}{9\cdot8}.
$$
Thus $k=9$, and
$$
\boxed{\widetilde\Lambda_2=9/T_{10},\qquad \operatorname{Var}(\widetilde\Lambda_2)=\lambda^2/8.}
$$
The <score function> is $10/\lambda-T_{10}$, so the <Fisher information> is $10/\lambda^2$. Its <Cramér-Rao lower bound> is $\lambda^2/10$, strictly below this estimator's <variance>.
Nevertheless, $9/T_{10}$ is <uniformly minimum-variance unbiased>. The observed $T_{10}$ is a <sufficient statistic>, and its family is complete: if $\mathbb E_\lambda h(T_{10})=0$ for all $\lambda>0$, then $\int_0^\infty h(t)t^9e^{-\lambda t}\,dt=0$ for all $\lambda>0$. Uniqueness of the <Laplace transform>, applied after an exponential tilt to make an <Lebesgue integrable function>, implies $h=0$ almost everywhere. The <Lehmann–Scheffé theorem> now proves the minimum-variance assertion. Failure to attain the Cramér–Rao bound does not disprove minimum <variance> within the class of <unbiased estimators>.
For the actual observations, the two <likelihood functions> are
$$
L_1(\lambda)=\frac{(3\lambda)^{10}}{10!}e^{-3\lambda},\qquad
L_2(\lambda)=\frac{\lambda^{10}3^9}{9!}e^{-3\lambda}.
$$
They differ by a factor independent of $\lambda$. The <likelihood principle> consequently requires the same inferential conclusions, since the observed data carry proportional likelihoods. But the two minimum-variance unbiased estimates are $\boxed{10/3\text{ and }3}$ respectively. Minimum-variance unbiased estimation therefore violates the <likelihood principle>: its unbiasedness and <variance> refer to the sampling plan, including the stopping rule, rather than solely to the observed <Likelihood function>.
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