Question

If we have \(n\) data points, what is the probability that a given data point does not appear in a bootstrap sample?

  1. \((1\) \(-\) \((1/n))^n\)
  2. \((1/n)^n\)
  3. \(1\) \(-\) \((1/n)^n\)

Answer

In bootstrap we randomly select \(n\) observations from a data set, with replacement i.e. a data point can be selected more than once. Bootstrap is a useful technique to find measures like variance, confidence intervals, prediction error etc associated with sampling estimates.

In the above question, we have \(n\) data points. The probability of selecting one point is therefore \(1/n\). The probability of not selecting that point is \(1\) \(-\) \((1/n)\). And this has to be repeated \(n\) times.

So, the probability a data point doesn’t appear in bootstrap sample is \((1\) \(-\) \((1/n))^n\).

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