ThmDex – An index of mathematical definitions, results, and conjectures.
Proof P3591 on R5273: Finiteness of expectation in general does not imply finiteness of variance for random real number
P3591
The first claim is a consequence of the results
(i) R5274: Probability mass partition for expectation of a random natural number
(ii) R268: Basel problem
(iii) R3414: Necessary and sufficient condition for convergence of real harmonic series

while the latter claim is a consequence of the results
(i) R5275: Probability mass partition for natural number moment of a random natural number
(ii) R2954: Standard real harmonic series diverges

$\square$