= Bootstrap sample
A <bootstrap sample> of size $n$ is drawn by <sampling with replacement> from the $n$ observed data slots. Some observations occur repeatedly. For a given slot, every draw misses it with <probability> $1-1/n$, so its omission <probability> after $n$ <independent> draws is
$$
\mathbb P_*(N_i=0)=\left(1-\frac1n\right)^n\longrightarrow e^{-1}.
$$
By <linearity of expectation>, the expected number of omitted slots is $n(1-1/n)^n$, whether or not their omission events are <independent>. For the three-observation <exact bootstrap distribution for three observations>, an individual slot is omitted with <probability> $8/27$ and the expected number of omitted slots is $8/9$. This finite-sample calculation explains both repeated values and incomplete coverage; $e^{-1}$ is a limiting expected proportion, not an exact fraction in every resample.
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