# Expected value of E(exp(1/X)) where X~N(0,sigma_x) [duplicate]

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

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.