Does $e^{c\mathrm{E}[\textbf{X}]} <= \mathrm{E}[e^{c\textbf{X}}]$ always hold, here $\textbf{X}$ is a random number and $c\in {\rm I\!R}$?
2026-04-01 06:30:28.1775025028
Proof of inequality about expectation
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Provided the expectations $E[e^{cX}]$ and $E[X]$ exist, by Jensen's inequality, we have $e^{c E[X]} \leq E[ e^{c X}]$ for any choice of $c \in \mathbb{R}$.