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 1:50Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question. 


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. 

