Discussion
Under Midzuno's sampling design the inclusion probability of \(i\)th element \(\pi_{i}\) is at least :
Answer: A
The inclusion probability is at least \(\dfrac{n - 1}{N - 1}\).
In Midzuno's scheme the first unit is drawn with probability proportional to size, \(p_{i}\), and the remaining \(n - 1\) units by SRSWOR from the other \(N - 1\). Unit \(i\) is included either as the first draw or as one of the later ones:
\[ \begin{aligned} \pi_{i} &= p_{i} + (1 - p_{i})\frac{n - 1}{N - 1} \\ &= \frac{n - 1}{N - 1} + \frac{N - n}{N - 1}\,p_{i} \end{aligned} \]
Since \(p_{i} \ge 0\), \(\pi_{i} \ge (n - 1)/(N - 1)\), with equality only for a unit of size zero.
\(n/N\) is the inclusion probability under SRSWOR, which is not a lower bound here (a small unit can fall below it). The forms \(n(n - 1)/(N - 1)\) and \(n(n - 1)/N\) are not probabilities in general, since they can exceed 1.
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