How does a Nakagami Random Variable behave?

Question

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$

I have two questions regarding this random variable,

1- Is a sum of Nakagami random variables also a Nakagami random variable? For example if I have $Z=aX+bY$ where $a,b$ are non-negative constants and $X,Y$ are Nakagami random variables, what would be the distribution of $Z$? Would it also be Nakagami? Is the mean of $Z$ equal to $a+b$?

2- Assume I have Z a Nakagami random variable, what would be the distribution of $|Z|$ and the distribution of $|Z|^2$?

Thanks

Answer 1

If $X$ is a Nakagami random variable with parameters $(m,\Omega)$ then $X\geqslant0$ almost surely and $mX^2=\Omega U_m$ where $U_m$ is a gamma random variable with parameter $m$.

This shows that the answer to every question in 1. is "No", that the first question in 2. is absurd and allows to find the answer to the second question in 2. since $E(U_m)=m$.