content_image/326761.jpg content_image/326764.jpg content_image/326765.jpg content_image/326766.jpg
When α < 0 and y ≤ µ , the limits of integration a = −∞ and b = y implies a ∗ = −∞ and b ∗ = y $^{∗}$, respectively. Substituting the values of a $^{∗}$, b∗ and κ in C$_{1}$, C$_{2}$ and C$_{3}$ and summing the expression yields, 

<img src='content_image/326761.jpg'>

Similarly, when α > 0 and y > µ , the limits of integration a = y and b = ∞ implies a ∗ = y ∗ and b ∗ = ∞ , respectively. Substituting the values of a $^{∗}$, b∗ and κ in C$_{1}$, C$_{2}$ and C$_{3}$ and evaluating the expression 1 − C$_{1}$ − C$_{2}$ − C$_{3}$ , yields 

<img src='content_image/326764.jpg'>

Combining the equations (A.10), (A.15) and (A.16), we have the cdf of the GAL distribution given by equation (3). ■ 

## Appendix A.3. Moment Generating Function 

Theorem 3 : Suppose Y ∼ GAL ( µ, σ, p, α ), then the mgf denoted M$_{Y}$ ( t ) is as follows, 

<img src='content_image/326765.jpg'>

Proof : Using the definition of the mgf we have, 

<img src='content_image/326766.jpg'>

Substituting the GAL pdf (2) into equation (A.18), leads to two cases depending on α > 0 or α < 0. We again use the notation κ = 2 p (1 − p ) /σ and break the region of integration 

