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Is there a way to construct a p.m.f. $f(X)$ that has an expected value $E(X)=\infty$ and $Pr(X=1)=0.5$?

For example, if you have $n$ boxes and $X$ measures the number of objects in each box, how would you find a function as described above? Is there a way?

Perhaps $f$ would have a heavy-tailed distribution?

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    For *finite* boxes the expected value will be finite as well (unless one box have $\infty$ many objects).2012-12-14
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    If $Pr(X=1) > 0$, you don't have a p.d.f. (probability **density** function), you have a p.m.f. (probability **mass** function).2012-12-14
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    Beat me to that. Just about to make the same comment on p.d.f. vs p.m.f..2012-12-14

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