Random variable with finite logarithmic first moment, infinite logarithmic variance

Asked Mar 04 '21 at 17:55

Active Mar 04 '21 at 18:16

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Could you provide an example of a random variable $X$ such that $|\mathbb{E}(\ln(X))|<\infty$ but $\text{Var}(\ln(X))=\infty$, if such a random variable exists at all?

edited Mar 04 '21 at 18:04

asked Mar 04 '21 at 17:55

Richard Hardy

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Let $Y$ be a positive variable with finite expectation and infinite variance, then set $X=\exp(Y).$ – whuber Mar 04 '21 at 18:31
Richard, you should perhaps include the condition that $Var(X)\leq \infty$. – Dayne Mar 05 '21 at 00:36
Sorry it should strict inequality above. – Dayne Mar 05 '21 at 04:25

Random variable with finite logarithmic first moment, infinite logarithmic variance

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