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

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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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". 


In addition to the Pareto distribution for suitable parameter values, the tdistribution 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. 

