If we have ##x_1, \ldots , x_n##, all distinct values, and then sample from this with replacement and thus obtain a bootstrap sample ##x^{\star}_1, \ldots , x^{\star}_n##, what is the probability that the bootstrap sample has only two unique values?(adsbygoogle = window.adsbygoogle || []).push({});

My attempt at a solution:

there are ##\binom{n}{2}## possible pairs in the original sample.

When sampling with replacement, there are ##n^n## possible bootstrap samples. The number of ways that two unique values can occur is ##n-1## so the sought-for probability is:

##\binom{n}{2} \frac{n-1}{n^n}##.

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# Boostrap sample probability question

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