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I would like to obtain the expected value of $ \mathbb{E}(\exp(1/X))$ where $X$ ~ $N(0,\sigma_x)$

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Because $$\exp(y) \gt \max(0,y)$$ for all numbers $y,$ $$\mathbb E[\exp(1/X)] \gt \mathbb E[1/\max(0,X)].$$ However, the latter is infinite because $X$ has a continuous nonzero density at $0.$ See I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that? for a full analysis.

Therefore $\mathbb E[\exp(1/X)]$ diverges; that is, it is infinite.

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