Gamma Distribution and Transformations

May 2010
6
0
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.
 
Nov 2009
517
130
Big Red, NY
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:

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

Uniqueness implies:

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

matheagle

MHF Hall of Honor
Feb 2009
2,763
1,146
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.
 
Nov 2009
517
130
Big Red, NY
\(\displaystyle \color{red}{\text{Not sure what you want.}}\)
\(\displaystyle \color{red}{\text{The guide is quite explicit.}}\)
\(\displaystyle \color{red}{\text{This has been solved here.}}\)
NOW, it's a Haiku.
 
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matheagle

MHF Hall of Honor
Feb 2009
2,763
1,146
if u say so
I'm not an expert on poetry