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Let $\lambda \in (0,\infty).$ Does there exists a minimizer for the set $$ \{ -\text{E}[X] + \lambda \text{Var}[X],\; X \in L^2(\Omega,\mathcal{F},P) \} ? $$

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The answer is no. By considering random variables $X$ taking only one real value $c$ and then letting $c\to\infty$, we see that the infimum of your set is $-\infty$ and hence not attained.

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