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shminux comments on Anticipating critical transitions - Less Wrong
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Similar weird things happen for the Cauchy distribution (whose probability density function is proportional to 1/(1+x^2)), which is symmetric around 0 but does not have mean 0 because the sum doesn't converge.
Exercise: what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?
From the same source:
Now I'm wondering if there is a symmetric distribution where the sample mean is a strictly worse estimator of the median than a single observation.