# Parameters of the Nakagami Distribution given a known Gamma distribution

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?

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.