▁ 0 , ▁L 0 ▁= ▁ 0 , ▁and ▁for ▁i ▁= ▁ 1 ▁to ▁n : ▁ ▁For ▁a ▁probability ▁density ▁function ▁f ( x ) ▁with ▁the ▁cum ulative ▁distribution ▁function ▁ ▁F ( x ), ▁the ▁Lor enz ▁curve ▁L ▁is ▁given ▁by : ▁ ▁where ▁ ▁denotes ▁the ▁average . ▁The ▁Lor enz ▁curve ▁L ( F ) ▁may ▁then ▁be ▁pl otted ▁as ▁a ▁function ▁paramet ric ▁in ▁x : ▁L ( x ) ▁vs . ▁F ( x ). ▁In ▁other ▁context s , ▁the ▁quantity ▁computed ▁here ▁is ▁known ▁as ▁the ▁length ▁bi ased ▁( or ▁size ▁bi ased ) ▁distribution ; ▁it ▁also ▁has ▁an ▁important ▁role ▁in ▁renew al ▁theory . ▁ ▁Alternatively , ▁for ▁a ▁cum ulative ▁distribution ▁function ▁F ( x ) ▁with ▁inverse ▁x ( F ), ▁the ▁Lor enz ▁curve ▁L ( F ) ▁is ▁directly ▁given ▁by : ▁ ▁The ▁inverse ▁x ( F ) ▁may ▁not ▁exist ▁because ▁the ▁cum ulative ▁distribution ▁function ▁has ▁intervals ▁of ▁constant ▁values . ▁▁ ▁However , ▁the ▁previous ▁formula ▁can ▁still ▁apply ▁by ▁general izing ▁the ▁definition ▁of ▁x ( F ): ▁x ( F 1 ) ▁= ▁inf ▁{ y ▁: ▁F ( y ) ▁ ≥ ▁F 1 } ▁ ▁For ▁an ▁example ▁of ▁a ▁Lor enz ▁curve , ▁see ▁P are to ▁distribution . ▁ ▁Properties ▁ ▁A ▁Lor enz ▁curve ▁always ▁starts ▁at ▁( 0 , 0 ) ▁and ▁ends ▁at ▁( 1 , 1 ). ▁