Math Help - Gamma Distribution and Transformations

1. Gamma Distribution and Transformations

Hi, I have the following question about Gamma Distributions and Transformations. Given the following:

It is easy to work out that the MGF of X is:

Now, my doubts lie upon the following. Could someone give me a detailed run through these next two questions?

As always, any help is greatly appreciated.

2. Originally Posted by aniguchisan
Hi, I have the following question about Gamma Distributions and Transformations. Given the following:

It is easy to work out that the MGF of X is:

Now, my doubts lie upon the following. Could someone give me a detailed run through these next two questions?

As always, any help is greatly appreciated.
(a) Multiplicative principle of MGF says:

$M_Z = M_{X_1}\cdot M_{X_2} = (1-\frac{r}{\beta})^{-\alpha_1}(1-\frac{r}{\beta})^{-\alpha_2} = (1-\frac{r}{\beta})^{-(\alpha_1+\alpha_2)}$

Uniqueness implies:

$Z \sim G\Big((\alpha_1+\alpha_2),\beta\Big)$

3. I'm not sure what you want.
The instructions are quite explicit.
This is not a difficult problem and it's been solved here several times.

4. Originally Posted by matheagle
$\color{red}{\text{Not sure what you want.}}$
$\color{red}{\text{The guide is quite explicit.}}$
$\color{red}{\text{This has been solved here.}}$
NOW, it's a Haiku.

5. if u say so
I'm not an expert on poetry