Editing Lorenz curve (section)

== Definition and calculation==
[[File:Gini coefficient US 2016.svg|upright=1.2|thumb|Lorenz curve for US wealth distribution in 2016 showing negative wealth and oligarchy]]
The Lorenz curve is a probability plot (a [[P–P plot]]) comparing the distribution of a [[Random variable|variable]] against a hypothetical uniform distribution of that variable. It can usually be represented by a function {{math|''L''(''F'')}}, where {{mvar|F}}, the cumulative portion of the population, is represented by the horizontal axis, and {{mvar|L}}, the cumulative portion of the total wealth or income, is represented by the vertical axis.

The curve {{mvar|L}} need not be a smoothly increasing function of {{mvar|F}}, For wealth distributions there may be oligarchies or people with negative wealth for instance.<ref>{{cite journal |last1=Li |first1=Jie |last2=Boghosian |first2=Bruce M. |last3=Li |first3=Chengli |title=The Affine Wealth Model: An agent-based model of asset exchange that allows for negative-wealth agents and its empirical validation |date=14 February 2018|arxiv=1604.02370v2 }}</ref>

For a discrete distribution of Y given by values {{math|''y''{{sub|1}}}}, ..., {{math|''y''{{sub|''n''}}}} in non-decreasing order {{math|(''y''{{sub|''i''}} ≤ ''y''{{sub|''i''+1}})}} and their probabilities <math>f(y_j) := \Pr(Y=y_j)</math> the Lorenz curve is the [[continuous function|continuous]] [[piecewise linear function]] connecting the points {{math|(''F''{{sub|''i''}}, ''L''{{sub|''i''}})}}, {{math|1=''i'' = 0 to ''n''}}, where {{math|1=''F''{{sub|0}} = 0}}, {{math|1=''L''{{sub|0}} = 0}}, and for {{math|1=''i'' = 1 to ''n''}}:
<math display="block">\begin{align}
F_i &:= \sum_{j=1}^i f(y_j) \\
S_i &:= \sum_{j=1}^i  f(y_j) \, y_j \\
L_i &:= \frac{S_i}{S_n}
\end{align}</math>

When all {{math|''y''{{sub|''i''}}}} are equally probable with probabilities {{math|1 / ''n''}} this simplifies to 
<math display="block">\begin{align}
F_i &= \frac i n \\
S_i &= \frac 1 n \sum_{j=1}^i \; y_j \\
L_i &= \frac{S_i}{S_n}
\end{align} </math>

For a [[continuous distribution]] with the [[probability density function]] {{mvar|f}} and the [[cumulative distribution function]]  {{mvar|F}}, the Lorenz curve {{mvar|L}} is given by:
<math display="block"> 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} </math>
where <math>\mu</math> denotes the average. The Lorenz curve {{math|''L''(''F'')}} may then be plotted as a function parametric in {{mvar|x}}: {{math|''L''(''x'')}} vs. {{math|''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.

Alternatively, for a [[cumulative distribution function]] {{math|''F''(''x'')}} with inverse {{math|''x''(''F'')}}, the Lorenz curve {{math|''L''(''F'')}} is directly given by:
<math display="block"> L(F) = \frac{\int_0^F x(F_1)\,dF_1}{\int_0^1 x(F_1)\,dF_1} </math>

The inverse {{math|''x''(''F'')}} may not exist because the cumulative distribution function has intervals of constant values.   However, the previous formula can still apply by generalizing the definition of {{math|''x''(''F'')}}:
<math display="block"> x(F_1) = \inf \{y : F(y) \geq F_1\}</math>
where {{math|inf}} is the [[infimum]].

For an example of a Lorenz curve, see [[Pareto distribution]].