Editing Gini coefficient (section)

=== Continuous probability distribution ===

When the population is large, the income distribution may be represented by a continuous [[probability density function]] ''f''(''x'') where ''f''(''x'') ''dx'' is the fraction of the population with wealth or income in the interval ''dx'' about ''x''.  If ''F''(''x'') is the [[cumulative distribution function]] for ''f''(''x''):

:<math>F(x)=\int_0^x f(x)\,dx</math>

and ''L''(''x'') is the Lorenz function:

:<math>L(x)=\frac{\int_0^x x\,f(x)\,dx}{\int_0^\infty x\,f(x)\,dx}</math>

then the [[Lorenz curve]] ''L''(''F'') may then be represented as a function parametric in ''L''(''x'') and ''F''(''x'') and the value of ''B'' can be found by [[integral|integration]]:

:<math>B = \int_0^1 L(F) \,dF. </math>

The Gini coefficient can also be calculated directly from the [[cumulative distribution function]] of the distribution ''F''(''y''). Defining ''μ'' as the mean of the distribution, then specifying that ''F''(''y'') is zero for all negative values, the Gini coefficient is given by:
:<math>G = 1 - \frac{1}{\mu}\int_0^\infty (1-F(y))^2 \,dy = \frac{1}{\mu}\int_0^\infty F(y)(1-F(y)) \,dy</math>
The latter result comes from [[integration by parts]]. ''(Note that this formula can be applied when there are negative values if the integration is taken from minus infinity to plus infinity.)''

The Gini coefficient may be expressed in terms of the [[quantile function]] ''Q''(''F'') ''(inverse of the cumulative distribution function: Q(F(x)) = x)''
: <math>G=\frac{1}{2 \mu}\int_0^1 \int_0^1 |Q(F_1)-Q(F_2)|\,dF_1\,dF_2 .</math>

Since the Gini coefficient is [[income inequality metrics|independent of scale]], if the distribution function can be expressed in the form ''f(x,&phi;,a,b,c...)'' where ''&phi;'' is a scale factor and ''a, b, c...'' are dimensionless parameters, then the Gini coefficient will be a function only of ''a, b, c...''.<ref name="McDonald1974">{{cite journal |last1=McDonald |first1=James B |last2=Jensen |first2=Bartell C. |date=December 1979 |title=An Analysis of Some Properties of Alternative Measures of Income Inequality Based on the Gamma Distribution Function |url= |journal=Journal of the American Statistical Association |volume=74 |issue=368 |pages=856–860 |doi= 10.1080/01621459.1979.10481042|access-date=}}</ref> For example, for the [[exponential distribution]], which is a function of only ''x'' and a scale parameter, the Gini coefficient is a constant, equal to 1/2.

For some functional forms, the Gini index can be calculated explicitly. For example, if ''y'' follows a [[log-normal distribution]] with the standard deviation of logs equal to <math>\sigma</math>, then <math>G = \operatorname{erf}\left(\frac{\sigma }{2 }\right)</math> where <math>\operatorname{erf}</math> is the [[error function]] ( since <math> G=2 \Phi \left(\frac{\sigma }{\sqrt{2}}\right)-1</math>, where  <math>\Phi</math> is the cumulative distribution function of a standard normal distribution).<ref name='LNdist'>Crow, E. L., & Shimizu, K. (Eds.). (1988). Lognormal distributions: Theory and applications (Vol. 88). New York: M. Dekker, page 11.</ref>  In the table below, some examples for probability density functions with support on <math>[0,\infty)</math> are shown. The Dirac delta distribution represents the case where everyone has the same wealth (or income); it implies no variations between incomes.{{fact|date=January 2025}}

:{| class="wikitable" style="float: left; margin-left: 1em;"
|-
! Income Distribution function !! PDF(x) !! Gini Coefficient
|-
| [[Dirac delta function]] || <math>\delta(x-x_0),\, x_0>0</math> || 0
|-
| [[Uniform distribution (continuous)|Uniform distribution]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Uniform Distribution |url=https://mathworld.wolfram.com/ |access-date=2022-11-30 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\begin{cases}
\frac{1}{b-a} & a\le x\le b \\ 0 & \mathrm{otherwise}
\end{cases}</math>
|| <math>\frac{(b-a)}{3(b+a)}</math>
|-
| [[Exponential distribution]]<ref>{{Cite web |title=Exponential Distribution {{!}} Definition {{!}} Memoryless Random Variable |url=https://www.probabilitycourse.com/chapter4/4_2_2_exponential.php |access-date=2022-11-30 |website=www.probabilitycourse.com}}</ref>
||<math>\lambda e^{-x\lambda},\,\,x>0</math>
||<math>1/2</math>
|-
| [[Log-normal distribution]]<ref name='LNdist'/><ref>For the log-normal with <math>\sigma</math> = 0, <math>\textrm{erf}(0)</math> = 0;  <math>2 \Phi(0)-1 = 2(0.5)-1</math> = 0.</ref>
||<math>\frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac 12 \left(\frac{\ln\,(x)-\mu}{\sigma}\right)^2}</math>
||<math>\textrm{erf}(\sigma/2)=2 \Phi \left(\frac{\sigma }{\sqrt{2}}\right)-1</math>
|-
| [[Pareto distribution]]<ref name="mathworld.wolfram.com">{{Cite web |title=Wolfram MathWorld: The Web's Most Extensive Mathematics Resource |url=https://mathworld.wolfram.com/ |access-date=2022-11-30 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\begin{cases}
\frac{\alpha k^\alpha}{x^{\alpha+1}} & x\ge k\\0 & x < k
\end{cases}</math>
||<math>\begin{cases}
1 & 0<\alpha < 1\\ \frac{1}{2\alpha -1} & \alpha \ge 1
\end{cases}</math>
|-
| [[Chi distribution]]<ref name="mathworld.wolfram.com"/>
||<math>f(x;k) = \begin{cases}
\dfrac{x^{k-1}e^{-x^2/2}}{2^{k/2-1}\Gamma\left(\frac{k}{2}\right)}, & x\geq 0 \\ 0, & x<0
\end{cases}
</math>
||<math>(-1)^k \left| I_{-1}(k,\tfrac{1}{2})\right|</math>
|-
| [[Chi-squared distribution]]<ref>{{Cite web |title=Chi-Squared Distribution -- from Wolfram MathWorld |url=https://mathworld.wolfram.com/Chi-SquaredDistribution.html |access-date=2023-01-11 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\frac{2^{-k/2} e^{-x/2} x^{k/2 - 1}}{\Gamma(k/2)}</math>
||<math>\frac{2\,\Gamma\left(\frac{1+k}{2}\right)}{k\,\Gamma(k/2)\sqrt{\pi}}</math>
|-
| [[Gamma distribution]]<ref name=McDonald1974/>
||<math>\frac{e^{-x/\theta}x^{k-1}\theta^{-k}}{\Gamma(k)}</math>
||<math>\frac{\Gamma\left(\frac{2k+1}{2}\right)}{k\,\Gamma(k)\sqrt{\pi}}</math>
|-
| [[Weibull distribution]]<ref>{{Cite web |title=Weibull Distribution: Characteristics of the Weibull Distribution |url=https://www.weibull.com/hotwire/issue14/relbasics14.htm |access-date=2022-11-30 |website=www.weibull.com}}</ref>
||<math>\frac {k} {\lambda}\, \left(\frac {x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k}</math>
||<math>1-2^{-1/k}</math>
|-
| [[Beta distribution]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Beta Distribution |url=https://mathworld.wolfram.com/ |access-date=2022-11-30 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\frac {x^{\alpha-1}(1-x)^{\beta-1}} {B(\alpha,\beta)}</math>
||<math>\left(\frac{2}{\alpha}\right)\frac{B(\alpha+\beta,\alpha+\beta)}{B(\alpha,\alpha)B(\beta,\beta)}</math>
|-
|[[Log-logistic distribution]]<ref>{{Cite web |title=The Log-Logistic Distribution |url=https://www.randomservices.org/random/special/LogLogistic.html |access-date=2022-11-30 |website=www.randomservices.org}}</ref>
|<math>\frac{ (\beta/\alpha)(x/\alpha)^{\beta-1} }
                       { \left (1+(x/\alpha)^{\beta} \right)^2  }</math>
|<math>1/\beta</math>
|}
{{clear}}

* <math>\Gamma(\,)</math> is the [[Gamma function]]
* <math>B(\,)</math> is the [[Beta function]]
* <math>I_k(\,)</math> is the [[Beta function|Regularized incomplete beta function]]