Bootstrap Coverage
Why does each bootstrap sample, on average, asymptotically, include about 63.2 percent of the original data?
Answer
When you draw \(n\) observations with replacement from a dataset of size \(n\), the probability that a specific point is never picked is
\[
\Bigl(1 - \tfrac1n\Bigr)^{n}
\;\approx\; e^{-1}
\;\approx\; 0.368.
\]
Therefore, the probability it is selected at least once is
\[
1 - \Bigl(1 - \tfrac1n\Bigr)^{n}
\;\approx\; 1 - e^{-1}
\;\approx\; 0.632,
\]