content_image/301213.jpg content_image/301215.jpg content_image/301216.jpg content_image/301217.jpg content_image/301218.jpg
<img src='content_image/301213.jpg'>

and when α < 0 and y ∗ > 0 (i.e., y > µ ) we have, 

<img src='content_image/301215.jpg'>

The integration of the terms M$_{5}$ to M$_{8}$ are similar to the case when α > 0 and results in the following mgf , 

<img src='content_image/301216.jpg'>

Combining the mgf for the two cases, i.e. equations (A.25) and (A.28), we have the mgf of the GAL distribution. ■ 

The mean, variance and skewness of the GAL distribution can be obtained from the GAL mgf (A.17). We state this in terms of a theorem below. 

Theorem 4 : Suppose Y ∼ GAL ( µ, σ, p, α ), then 

<img src='content_image/301217.jpg'>

where E ( Y ), V ( Y ) and S ( Y ) denote the mean, variance and skewness, respectively. Proof : Taking logarithm of the mgf and keeping terms involving t we have, 

<img src='content_image/301218.jpg'>

