How many distinct points appear in a bootstrap sample?

Question

You have n = 500 observations and draw a bootstrap sample of size 500 with replacement. Some originals appear several times, others not at all. About how many distinct original observations do you expect to appear at least once in the bootstrap sample?

Accepted Answer

A specific observation is included at least once with probability 1 - \left(1 - 	frac{1}{n}ight)^{n}. For large n this is about 1 - e^{-1} \approx 0.632. Expected number of distinct points. By linearity of expectation, add that inclusion probability over all n points: n \left[ 1 - \left(1 - \frac{1}{n}ight)^{n} ight] = 500 	imes (1 - 0.3675) \approx 500 	imes 0.6325 \approx 316. So a bootstrap sample of 500 draws contains only about 316 distinct originals; the rest of the "slots" are duplicates. This is why a single bootstrap sample sees roughly two-thirds of the data. Takeaway. Beca…

How many distinct points appear in a bootstrap sample?

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