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According to the material I have in hand for Bayesian Econometrics, we define the pdf of a Gamma-2 distributed random variable $Z$ with parameter $\mu > 0$ and degrees of freedom $\nu > 0 $, that is, $Z \sim G2(\mu,\nu)$ is given by

$p(z|\mu,\nu)=c^{-1}z^{\frac{\nu-2}{2}}\exp(\frac{-z\mu}{2})$

where $c = (\frac{2}{\mu})^{\frac{\nu}{2}}\Gamma(\frac{\nu}{2})$

It holds that $\mu Z \sim \chi^{2}(\nu)$ and $Z \sim G(\mu/2,\nu/2)$ (Gamma Distribution).

My problems are the following:

1.- What does 2, in Gamma-2, mean? Is it something with parameters are equal to 2?

2.- What is the relationship between a Gamma-2 and a Gamma distribution $f(x|\alpha,\beta) = \frac{1}{\Gamma(\alpha,\beta)}x^{\alpha-1}e^{-\beta x}$

3.- I cannot see the relation with an uninformative prior for a Gaussian model such that $p(\sigma^2) \propto \sigma^{-2}$.

I'd appreciate some clarification on these points as I got stuck in the derivations and cannot see why a Gamma-2 is used.

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