Recall that the density of a gamma distribution with shape parameter \(\alpha\) and scale parameter \(\theta\) has a density given by \(\mathrm{f}(y)=\) \(\left[\theta^{\alpha} \Gamma(\alpha)\right]^{-1} y^{\alpha-1} \exp (-y / \theta)\) and \(k\) th moment given by \(\mathrm{E}\left(y^{k}\right)=\) \(\theta^{k} \Gamma(\alpha+k) / \Gamma(\alpha)\), for \(k>-\alpha\).

a. For the GB2 distribution, show that

\[\mathrm{E}(y)=e^{\mu} \frac{\Gamma\left(\alpha_{1}+\sigma\right) \Gamma\left(\alpha_{2}-\sigma\right)}{\Gamma\left(\alpha_{1}\right) \Gamma\left(\alpha_{2}\right)} .\]

b. For the generalized gamma distribution, show that

\[\mathrm{E}(y)=e^{\mu} \Gamma\left(\alpha_{1}+\sigma\right) / \Gamma\left(\alpha_{1}\right) .\]

c. Calculate the moments of a Burr type 12 density.