# What is the distribution of these functions of Nakagami random variables?

I am new to this forum and hope I can get help.

A Nakagami random variable $X$ with parameter $m$ has the following pdf

$$X\sim \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$

Define $$T:=|X|^2$$

1- Is T a normalized Gamma random variable with parameter m? Why is it called Normalized? What is the pdf?

2- What would change if we define a function as the following $$W:= c |X|^2 \sim ?$$ where c is non-negative constant. Would $W$ also be Gamma distributed - if the answer to the above question is yes - but what would change, does the parameter change?

• Very closely related: stats.stackexchange.com/questions/72479/…. More about sums of gamma variables can be found with a site search. – whuber Nov 4 '14 at 21:40
• Thanks for pointing me to this search engine. I noticed the similarity. However I couldn't find the answer to many of my questions such as what does normalized gamma mean? What happens when you multiply the random variable by constant does the mean change or does the parameter change? – Tyrone Nov 4 '14 at 21:49
• You are asking a lot of different questions, so no one thread will answer them all. The ones you just mentioned are so basic that (a) there will be many answers but (b) they will be hard to search for. You could try threads in which "scale" is a prominent word. It might help to notice that multiplying any quantity by a (nonnegative) constant merely changes its units of measurement. As far as "normalized Gamma" goes, that could mean many things, so you ought to supply a reference or some context to help readers figure out the intended meaning. – whuber Nov 4 '14 at 22:20
• I understand what you mean, it is also confusing in my head that is why I posed so many questions. The term normalized that I am asking for, I found it in a wireless communication paper where a link has a Nakagami fading with parameter $N$, they claim that the magnitude square of this random variable has "normalized Gamma" distribution – Tyrone Nov 4 '14 at 22:33
• The reason I am asking for a scaled random variable is because I dont see or understand how it will change my distribution in this case the Gamma distribution. – Tyrone Nov 4 '14 at 22:34