Does such a distribution exist? The Cauchy distribution has infinite variance but its mean is also undefined.
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2$\begingroup$ Look up "power law": en.wikipedia.org/wiki/Power_law . $\endgroup$– Joseph O'RourkeSep 19, 2011 at 0:19
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$\begingroup$ This is more suitable for math.stackexchange.com than here. $\endgroup$– Deane YangSep 19, 2011 at 0:57
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$\begingroup$ Math questions posted here are supposed to be research questions, whereas this is more the sort of thing found in textbooks. I suspect you'll be back with more questions of the former kind. $\endgroup$– Michael HardySep 19, 2011 at 5:49
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2 Answers
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Consider the density function $f(x)= (3/2) x^{-5/2}$ on the interval from $1$ to infinity.
More generally, Google for the term "Pareto distribution".
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In addition to the Pareto distribution for suitable parameter values, the t-distribution has a mean of 0 if the number of degrees of freedom is more than 1, but has infinite variance if it is not more than 2.