Say I re-sample $N$ items with replacement from a numbered item sample of size $N$. What is the average number of data items that are not selected in each such sampling?
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Hint: what is the probability that a specific item is not sampled in $1$ draw? what is the probability that it is not sampled in $N$ draws? Now use the linearity of expectation.
Ross Millikan
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If you could elaborate... the probability that a specific item is not sampled in $N$ draws is $[(N-1)/N]^N$. – Omri Feb 12 '13 at 21:21
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@Omri: so the average number you would expect not to be sampled at all is $N\left( \frac {N-1}N \right)^N$. – Ross Millikan Feb 12 '13 at 21:43
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Thanks! Interestingly, the average portion of $N$ that is not sampled approaches $1/e$ as $N$ goes to infinity... – Omri Feb 12 '13 at 22:18