Lets say we have a sample $\mathbf{X} = \{x_1, x_2, \dots, x_N\}$. We draw $N$ points from $\mathbf{X}$ with replacement (do a $\textit{bootstrap})$. What is the expectation of size of bootstrapped sample if we consider replicated points as one entity?
It seems that it goes to $N\big(1-\frac{1}{e}\big)$ as $N$ goes to infinity. But the answer is not the same for small samples.
Each point has probability $1-(1-1/N)^N$ of being drawn. (This goes to $1-1/\mathrm e$ for $N\to\infty$.) Thus by linearity of expectation, the expected number of points drawn is $N\left(1-(1-1/N)^N\right)$.