# For a normal distributed random variable X what is the distribution of c/X

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

• This seems to be called a reciprocal normal distribution, not sure if that helps? Oct 14, 2023 at 10:56
• 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. Oct 14, 2023 at 11:15
• The claim in your last sentence is mistaken. The CLT doesn't tell you that Oct 14, 2023 at 14:30
• 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. Oct 14, 2023 at 22:40
• 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 Oct 14, 2023 at 22:49