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How do you prove that the sample median of $X_i, \ldots, X_n$ where $i$ is odd converges to the median of U$(0,1)$? Do you have to find the cdf of the median, find the p.m.f. of median, and then show that as $n$ increases, the variance of the p.m.f. gets smaller, eventually becoming a point mass around the true median?

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