F ), ▁then ▁− X ▁has ▁the ▁Lor enz ▁curve : ▁ ▁L ▁− ▁X ▁ ▁= ▁ 1 ▁− ▁L ▁X ▁( 1   −   F ) ▁ ▁The ▁Lor enz ▁curve ▁is ▁changed ▁by ▁transl ations ▁so ▁that ▁the ▁equality ▁gap ▁F   −   L ( F ) ▁changes ▁in ▁proportion ▁to ▁the ▁ratio ▁of ▁the ▁original ▁and ▁translated ▁means . ▁ ▁If ▁X ▁is ▁a ▁random ▁variable ▁with ▁a ▁Lor enz ▁curve ▁L ▁X ▁( F ) ▁and ▁mean ▁ μ ▁X ▁, ▁then ▁for ▁any ▁constant ▁c ▁ ≠ ▁− μ ▁X ▁, ▁X ▁+ ▁c ▁has ▁a ▁Lor enz ▁curve ▁defined ▁by : ▁ ▁For ▁a ▁cum ulative ▁distribution ▁function ▁F ( x ) ▁with ▁mean ▁ μ ▁and ▁( general ized ) ▁inverse ▁x ( F ), ▁then ▁for ▁any ▁F ▁with ▁ 0 ▁< ▁F ▁< ▁ 1 ▁: ▁If ▁the ▁Lor enz ▁curve ▁is ▁differenti able : ▁ ▁If ▁the ▁Lor enz ▁curve ▁is ▁twice ▁differenti able , ▁then ▁the ▁probability ▁density ▁function ▁f ( x ) ▁exists ▁at ▁that ▁point ▁and : ▁ ▁If ▁L ( F ) ▁is ▁continu ously ▁differenti able , ▁then ▁the ▁tang ent ▁of ▁L ( F ) ▁is ▁parallel ▁to ▁the ▁line ▁of ▁perfect ▁equality ▁at ▁the ▁point ▁F ( μ ). ▁ ▁This ▁is ▁also ▁the ▁point ▁at ▁which ▁the ▁equality ▁gap ▁F   −   L ( F ), ▁the ▁vertical ▁distance ▁between ▁the ▁Lor enz ▁curve ▁and ▁the ▁line ▁of ▁perfect ▁equality ,