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I have run into the following expectation and am not sure what it is.

$ X \sim N(0, \sigma^2)$

Then $\mathrm{e}^{x}$ is lognormal, but what is the expectation of $\mathbb{E}\left[\exp(X^2)\right]$? $X^2$ is a chi squared distribution, but then what is the distribution of $\exp(X^2)$?

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