Expected number of times a point appears in a bootstrap sample

Question

You have n observations and draw a bootstrap sample of size n with replacement. Focus on one particular observation. Across the n draws it might be picked zero times, once, or several times. What is the expected number of times that particular observation appears in the bootstrap sample?

Accepted Answer

Each of the n draws selects this specific point with probability 	frac{1}{n}, independently. The count of appearances is therefore a sum of n independent indicators, one per draw. Expected count. By linearity of expectation, n 	imes \frac{1}{n} = 1. So on average each observation appears exactly once, which makes sense: the resample has n slots spread over n points. The distribution is roughly Poisson with mean 1: about 37\% of points appear zero times, about 37\% once, and the remaining quarter two or more times. Takeaway. In a size-n bootstrap sample, every observation appears exactly onc…

Expected number of times a point appears in a bootstrap sample

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