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?