this occurs for all s ∈ ( y $^{∗}$/α, ∞ ). So we have, f ( y | θ ) = κ ∫ y ∗ α 0 1 √$_{2}$$_{π}$ exp { − 1 2 [ s 2 + 2( y ∗ − αs )( p − 1) ] } ds + κ ∞ y ∗ α 1 √$_{2}$$_{π}$ exp − 1 2 s 2 + 2( y ∗ − αs ) p ] } ds = κ ∫ y ∗ α 0 1 √$_{2}$$_{π}$ exp { − 1 2 [ s 2 + 2( y ∗ − αs ) p$_{α}$$_{−}$ ] } ds + κ ∞ y ∗ α 1 √$_{2}$$_{π}$ exp − 1 2 s 2 + 2( y ∗ − αs ) p$_{α}$$_{+}$ ds = κ [ Φ ( y ∗ α − αp$_{α}$$_{−}$ ) − Φ( − αp$_{α}$$_{−}$ ) ] exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ } + κ Φ αp$_{α}$$_{+}$ − y ∗ α )] exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 α $^{2}$p2 α + } , (A.8) where the integration details are similar to Case (ii) and have been suppressed to avoid monotonicity and save space. Combining all the four cases, i.e. equations (A.5) to (A.8), we have the pdf of the GAL distribution given by equation (2). ■ Theorem 2 : Suppose Y ∼ GAL ( µ, σ, p, α ) and let y ∗ = ( y − µ ) /σ , then the cdf F is, Appendix A.2. Cumulative Distribution Function F ( y | θ ) = ( 1 − 2Φ ( − y ∗ | α $_{|}$) − 2 p (1 − p ) p$_{α}$$_{−}$ exp { − y $^{∗}$p$_{α}$$_{−}$ + 1 2 α $^{2}$p2 α$_{−}$ } [ Φ ( y ∗ α − αp$_{α}$$_{−}$ ) − Φ( − αp$_{α}$$_{−}$ ) ] ) I ( y ∗ α > 0 ) + I ( α < 0) − 2 p (1 − p ) p$_{α}$$_{+}$ exp { − y $^{∗}$p$_{α}$$_{+}$ + 1 2 α $^{2}$p2 α + } × Φ [ αp$_{α}$$_{+}$ − y ∗ α I ( y ∗ α > 0 )] . (A.9) Proof : We note that for any cdf F ( y | θ ) = y ∫ −∞ f ( v | θ ) dv = 1 − ∞ ∫ y f ( v | θ ) dv . This property is used in deriving the cdf when y > µ to avoid breaking the region of integration as ( −∞ , µ ) ⋃ ( µ, y ). We let v ∗ = ( v − µ ) /σ , combine cases and derive as follows. 30