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Hopefully an easy one. My apologies for not using math format, I am not sure how to do so.

I have a known Gamma distribution f(Y), say shape=3 and scale=2. I also know that that the distribution of f(Y)^1/2, i.e. the square root of f(Y), can be described as a Nakagami distribution f(X) where X=sqrt(Y). What I'm after is how to identify the parameters of that Nakagami distribution? I have found links describing how to go the other way (e.g https://handwiki.org/wiki/Nakagami_distribution) but it seems like the Omega parameter in that link is unknown in my case.

Through simulation and recovery I believe the answer is that the Nakagami shape parameter m is equal to the Gamma shape, and the Nakagami scale parameter is equal to the Gamma shape*scale. In my example, the Nakagami distribution would have a shape of 3, and a scale of 6. If this is true, mathematically, could anyone point me to a reference?

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Per Wikipedia, given a gamma distributed $Y$ with shape $k$ and scale $\theta$, $X=\sqrt{Y}$ is Nakagami with parameters $m$ and $\Omega$, where $$ k=m \quad\text{and}\quad\theta=\frac{\Omega}{m}$$ or $$ m=k \quad\text{and}\quad\Omega=\theta m.$$ If Wikipedia is not trustworthy enough to you, you may want to dig through the references given there.

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  • $\begingroup$ Oh wow, somehow I didn't make the connection between the two because I was stuck on what I "know" from the Gamma side. Thank you Stephen. $\endgroup$
    – Howchie
    Jul 29, 2022 at 0:40

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