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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.

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  • $\begingroup$ If you could elaborate... the probability that a specific item is not sampled in $N$ draws is $[(N-1)/N]^N$. $\endgroup$
    – Omri
    Feb 12, 2013 at 21:21
  • $\begingroup$ @Omri: so the average number you would expect not to be sampled at all is $N\left( \frac {N-1}N \right)^N$. $\endgroup$ Feb 12, 2013 at 21:43
  • $\begingroup$ Thanks! Interestingly, the average portion of $N$ that is not sampled approaches $1/e$ as $N$ goes to infinity... $\endgroup$
    – Omri
    Feb 12, 2013 at 22:18

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