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The nonzero leading coefficients and distinct degrees make $P_0,\dots ,P_{n-1}$ a basis of the polynomials of degree at most $n-1$. Consequently the given orthogonality implies ${\int }_a^{b}f(x)Q(x)dx=0$ for every such polynomial $Q$.

If $f$ has infinitely many distinct zeros in $(a,b)$, there is nothing to prove. Otherwise suppose it has fewer than $n$. Among those zeros, list the sign-changing ones as $t_1<\cdots <t_m$, with $m<n$, and put $Q(x)={\prod }_{j=1}^m(x-t_j)$, using $Q=1$ when $m=0$. On each interval between zeros, $f$ has a fixed sign by continuity. Crossing a sign-changing zero flips both $f$ and $Q$; crossing any other zero flips neither sign. Thus $fQ$ has one constant sign wherever nonzero. Since $f$ is not identically zero, continuity gives an open interval on which $fQ$ is strictly of that sign. Hence its integral is nonzero, contrary to orthogonality to $Q$. Therefore $f$ has at least $n$ distinct zeros in $(a,b)$. This is the reusable orthogonality forces many interior zeros argument.

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 givesThus $k=9$, andThe 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 ${\int }_0^{\infty }h(t)t^9{e}^{-\lambda t}dt=0$ 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 areThey 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.

The dominated convergence theorem states that if measurable functions converge almost everywhere and their absolute values are bounded by a single Lebesgue integrable function, their limit is integrable and their integrals converge to its integral.

Put $b_j=a_j/j$. The hypothesis makes $b_j$ summable. On the positive integers with counting measure define $f_n(j)=(j/n)b_j{1}_{j\le n}$. For each fixed $j$ it tends to zero and $0\le f_n(j)\le b_j$. Dominated convergence givesthe stated Kronecker lemma.

For a bounded interval union $B$, $S_j\sim N(0,j)$ has density bounded by $1/\sqrt{2\pi j}$. Hence $\mathbb{P}(S_j\in B)\le |B|/\sqrt{2\pi j}$, andBy Tonelli's theorem, the nonnegative series is finite almost surely. Kronecker lemma then makes the empirical occupation fraction of $B$ tend to zero. The family of finite rational-endpoint interval unions is countable, so intersecting their probability-one events proves simultaneous convergence for all of them. For the whole line, every indicator is one. ThusThere is no countably additive Borel measure extending these values: $\mathbb{R}={\bigcup }_{n\ge 1}(-n,n)$ would have measure zero by continuity from below, contrary to its assigned measure one. The escaped empirical mass cannot be captured by a probability measure on the real line.

This is the Gaussian random walk spends zero density of time in bounded sets phenomenon.