M$_{6}$ = − κ ∫ µ $_{−∞}$exp { ty } Φ( − αp$_{α}$$_{−}$ ) exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 ( αp$_{α}$$_{−}$ ) 2 } dy, (A.26) M$_{7}$ = κ ∫ µ $_{−∞}$exp { ty } Φ ( αp$_{α}$$_{+}$ − y ∗ α ) exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 ( αp$_{α}$$_{+}$ ) 2 } dy, and when α < 0 and y ∗ > 0 (i.e., y > µ ) we have, M$_{8}$ = κ ∫ ∞ µ exp { ty } Φ ( αp$_{α}$$_{+}$ ) exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 ( αp$_{α}$$_{+}$ ) 2 } dy. (A.27) M$_{Y}$ ( t ) = 2 p (1 − p ) [ ( p$_{α}$$_{+}$ − p$_{α}$$_{−}$ ) ( p$_{α}$$_{−}$ − σt )( p$_{α}$$_{+}$ − σt ) ] exp { µt + 1 2 α $^{2}$σ$^{2}$t2 } Φ( − ασt ) . (A.28) E ( Y ) = µ + 2 | α | σ √$_{2}$$_{π}$ + σ (1 − 2 p ) p (1 p ) V ( Y ) = α $^{2}$σ2 ( 1 − 2 π ) + σ 2 [ 1 − 2 p + 2 p 2 p $^{2}$(1 − p ) 2 ] S ( Y ) = α 3 √ 2 π ( 4 π − 1 ) + 2 [ (1 − p ) $^{3}$− p 3 p $^{3}$(1 − p ) 3 ] { α 2 ( 1 − 2 π ) + [ 1 − 2 p +2 p 2 p $^{2}$(1 − p ) 2 ]} 3 / 2 (A.29) ln M$_{Y}$ ( t ) ∝ − ln( p$_{α}$$_{−}$ − σt ) − ln( p$_{α}$$_{+}$ − σt ) + µt + 1 2 α $^{2}$σ$^{2}$t2 + ln Φ( | α | σt ) . (A.30) 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, The integration of the terms M$_{5}$ to M$_{8}$ are similar to the case when α > 0 and results in the following mgf , 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 36