= Solution
{c}
The <log-likelihood function> for the <Laplace distribution> is
$$
\ell(\kappa)=-n\log2-\sum_{i=1}^n|Y_i-\kappa|.
$$
Thus <maximum likelihood estimation> is equivalent to minimizing the sum of <absolute values>. Write $n=2m+1$ and sort the data as $Y_{(1)}\le\cdots\le Y_{(n)}$. Between observations, if $r$ values lie to the left of $\kappa$, the <derivative> of $\sum_i|Y_i-\kappa|$ is $r-(n-r)=2r-n$. It is negative before the middle observation and positive after it. The minimum occurs at the <sample median>:
$$
\boxed{\widehat\kappa=Y_{(m+1)}=Y_{((n+1)/2)}.}
$$
This minimizer remains unique for odd $n$ even when some observations coincide, since fewer than half the observations can lie strictly on either side of the middle value.
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