problem 39

4.3.2 problem 39

Let $I_{j, 1}$ and $I_{j, 2}$ be the indicator random variables for the $j$ -th person being sampled by the first and second researchers respectively.

$P (I_{j, 1} = 1) = \frac{(\frac{N - 1}{m - 1})}{(\frac{N}{m})}$ . $P (I_{j, 2} = 1) = \frac{(\frac{N - 1}{n - 1})}{(\frac{N}{n})}$ . Since sampling is done independently by the two researchers, $P (I_{j, 1} = 1, I_{j, 2} = 1) = \frac{(\frac{N - 1}{m - 1}) (\frac{N - 1}{n - 1})}{(\frac{N}{m}) (\frac{N}{n})}$ .

Let $X = \sum_{j = 1}^{n} (I_{j, 1} I_{j, 2})$ be the number of people sampled by both researchers. Then,

E (X) = E (n \sum j = 1 (I_{j, 1} I_{j, 2})) = n \sum j = 1 E (I_{j, 1} I_{j, 2}) = n \sum j = 1 \frac{(\frac{N - 1}{m - 1}) (\frac{N - 1}{n - 1})}{(\frac{N}{m}) (\frac{N}{n})} = n \frac{(\frac{N - 1}{m - 1}) (\frac{N - 1}{n - 1})}{(\frac{N}{m}) (\frac{N}{n})}

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