When is the approximation $\mathbb{E}\left\{{e^{f(x)}}\right\} = e^{\mathbb{E}\left\{f(x)\right\}}$ tight? I mean for which function of f(x) and which conditions? there is any theorem on it?
2026-03-27 13:46:12.1774619172
question on expected value of exponential functions
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If you've heard of Jensen's inequality, then using that, LHS is greater than or equal to RHS. Equality will occur iff $f(x)$ is a constant with probability 1. I'm assuming of course that the expectation on left is finite.