Past exam of the computer science course of the University of Oxford / 2015 / Probability and computing / 1 / e / Solution

Choose a positive integer-valued random variable with probability mass functionThese masses are positive. They sum to one because the partial fraction decompositionmakes their sum a telescoping series with value $2/(1\cdot 2)=1$. Its expectation is another telescoping series:By contrast, since $k+1\le 2k$ and $k+2\le 3k$ for $k\ge 1$, its second moment satisfiesThe comparison uses the divergent harmonic series. To check the variance directly, for $k\ge 4$ we have $(k-2{)}^2\ge k^2/4$, so the corresponding tail of $\mathbb{E}[(X-2{)}^2]$ also diverges. ThusHere “unbounded” means infinite variance for this single probability distribution.