Discussion
Consider a population of 50 units, {1, 2, 3, ....., 50} and suppose that 50 possible samples are listed as {1}, {1, 2}, {1, 2, 3}, ....., {1, 2, 3, ....., 50}. One of these samples is choosen at random. Let \(\pi_{i}\) be the probability that unit i of the population is in the selected sample. Then which of the following is necessarily true ?
Answer: A
The necessarily true statement is that the expected sample size is 25.5.
The 50 samples have sizes 1, 2, ..., 50, each chosen with probability 1/50, so
\[ \begin{aligned} E(\text{size}) &= \frac{1 + 2 + \ldots + 50}{50} \\ &= \frac{1275}{50} \\ &= 25.5 \end{aligned} \]
The inclusion probabilities always add up to the expected sample size, \(\sum \pi_{i} = E(n) = 25.5\), so neither 1 nor 25 is right for the sum. (Directly, unit i is in the samples of size i or more, so \(\pi_{i} = (51 - i)/50\), and these add to 25.5.)
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