Assume $X\sim \mathcal{N}(\mu, \sigma^2)$

For a normal distributed random variable $X,$ what is the distribution of $c/X$?

I had a look at ratio distributions but could not find it.

PS: The issue originally was raised when I asked for the distribution of $1/\hat{E}[Z]$. I know from the CLT that the arithmetic mean $\hat{E}[Z]=1/n \sum_{i=1}^n Z_i$ is (tends to be) normally distributed, for "suitably" distributed $Z_i$ and sufficiently big sample sizes

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    $\begingroup$ This seems to be called a reciprocal normal distribution, not sure if that helps? $\endgroup$
    – PBulls
    Oct 14, 2023 at 10:56
  • $\begingroup$ Thank you a lot! It is obviously true if I think about the delta transformation method. Feel free to post an answer and I will accept it. $\endgroup$
    – Ggjj11
    Oct 14, 2023 at 11:15
  • $\begingroup$ The claim in your last sentence is mistaken. The CLT doesn't tell you that $\endgroup$
    – Glen_b
    Oct 14, 2023 at 14:30
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    $\begingroup$ Unless you start at the normal, it is not actually normal at any $n$. You assert that it is on your post. It approaches the normal in the limit. It doesn't get there at any same size. In many situations this may not be a big issue practically as long as we don't assert that it is normal. $\endgroup$
    – Glen_b
    Oct 14, 2023 at 22:40
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    $\begingroup$ Specifically, $\bar{Z}$ is not actually normally distributed if the $Z_i$ aren't. Approximately, sure in the right circumstances, albeit you don't usually know if you're in them $\endgroup$
    – Glen_b
    Oct 14, 2023 at 22:49

1 Answer 1


This seems to be called the reciprocal normal distribution.


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