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, F ( y | θ ) = 2Φ ( − y ∗ α ) − 2 p (1 − p ) p$_{α}$$_{−}$ exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ } [ Φ ( y ∗ α − αp$_{α}$$_{−}$ ) − Φ( − αp$_{α}$$_{−}$ ) ] − 2 p (1 − p ) p$_{α}$$_{+}$ exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 α $^{2}$p2 α + } Φ ( αp$_{α}$$_{+}$ − y ∗ α ) . (A.15) Similarly, when α > 0 and y > µ , the limits of integration a = y and b = ∞ implies a ∗ = y ∗ and b ∗ = ∞ , respectively. Substituting the values of a $^{∗}$, b∗ and κ in C$_{1}$, C$_{2}$ and C$_{3}$ and evaluating the expression 1 − C$_{1}$ − C$_{2}$ − C$_{3}$ , yields Combining the equations (A.10), (A.15) and (A.16), we have the cdf of the GAL distribution given by equation (3). ■ F ( y | θ ) = 1 − 2Φ ( − y ∗ α ) − 2 p (1 − p ) p$_{α}$$_{−}$ exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ } [ Φ ( y ∗ α − αp$_{α}$$_{−}$ ) − Φ( − αp$_{α}$$_{−}$ ) ] − 2 p (1 − p ) p$_{α}$$_{+}$ exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 α $^{2}$p2 α + } Φ ( αp$_{α}$$_{+}$ − y ∗ α ) . (A.16) M$_{Y}$ ( t ) = 2 p (1 − p ) [ ( p$_{α}$$_{+}$ − p$_{α}$$_{−}$ ) ( p$_{α}$$_{−}$ − σt )( p$_{α}$$_{+}$ − σt ) ] exp { µt + 1 2 α $^{2}$σ$^{2}$t2 } Φ ( | α | σt ) . (A.17) M$_{Y}$ ( t ) = ∫ ∞ $_{−∞}$exp( ty ) f ( y | µ, σ, p, α ) dy. (A.18) 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, Proof : Using the definition of the mgf we have, 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 33