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Does such a distribution exist? The Cauchy distribution has infinite variance but its mean is also undefined.

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closed as off topic by Andreas Blass, Gerald Edgar, Deane Yang, Will Jagy, George Lowther Sep 19 '11 at 1:50

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Look up "power law": . – Joseph O'Rourke Sep 19 '11 at 0:19
This is more suitable for than here. – Deane Yang Sep 19 '11 at 0:57
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. – Michael Hardy Sep 19 '11 at 5:49

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.

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