= Solution
{c}
With immediate replacement and uniform mixing, each capture is a Bernoulli trial with tagged probability $p=k/N$. Counting failures before the $r$th tagged capture gives the stated <negative binomial distribution>. Independent runs on the successive days give the sample model.
Writing $S=\sum_i x_i$, the log-<likelihood> is $\ell(p)=nr\log p+S\log(1-p)+\text{constant}$. Its derivative vanishes at
$$
\boxed{\widehat p=\frac{nr}{nr+S}=\frac r{r+\overline x}.}
$$
The second derivative is negative. If $S=0$, this formula gives the boundary maximum $p=1$; on the strict physical parameter space $p<1$, it is a supremum rather than an attained interior maximum.
Using the supplied negative-binomial <mean>, $\mathbb ES=nr(1-p)/p$. Thus total <Fisher information> for the sample is
$$
\boxed{I_n(p)=-\mathbb E\ell''(p)=\frac{nr}{p^2}+\frac{nr}{p(1-p)}=\frac{nr}{p^2(1-p)}.}
$$
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