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If X is a normal distribution $N(0,\sigma^2)$ is $\frac{1}{X}$ any sort of "official" distribution or something that should just be computed?

In particular I'm looking to find the expectation $E[\frac{1}{X}]$ where X is a Brownian motion.

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    It has a name (Recinormal Distribution!) and comes up in some applications.2012-12-01
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    @AndréNicolas, I filled in your rain example comment. So it did not go to waste. Even though it it gone forever.2012-12-01
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    @AndréNicolas Interesting. Can it have a finite mean and variance?2012-12-01
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    @Dirk Are you reading the answers to your questions?2012-12-01
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    @did I am I was just wondering if there was some exceptional cases involving this Recinormal Distribution2012-12-01

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As I answered to your previous question, $E[1/X]=\infty$ for a normal distribution.