structural:scholar_edition
Definition and calculation

For a continuous distribution with the probability density function f and the cumulative distribution function F, the Lorenz curve L is given by: L (F (x)) = ∫ − ∞ x t f (t) d t ∫ − ∞ ∞ t f (t) d t = ∫ − ∞ x t f (t) d t μ {\displaystyle L(F(x))={\frac {\int _{-\infty }^{x}t\,f(t)\,dt}{\int _{-\infty }^{\infty }t\,f(t)\,dt}}={\frac {\int _{-\infty }^{x}t\,f(t)\,dt}{\mu }}} where μ {\displaystyle \mu } denotes the average. The Lorenz curve L (F) may then be plotted as a function parametric in x: L (x) vs. F (x). In other contexts, the quantity computed here is known as the length biased (or size biased) distribution; it also has an important role in renewal theory.