Editing Gini coefficient (section)

=== Discrete probability distribution ===

For a [[discrete probability distribution]] with probability mass function <math>f ( y_i ),</math> <math qid=Q120636410>i = 1,\ldots, n</math>, where  <math>f ( y_i )</math> is the fraction of the population with income or wealth <math>y_i >0 </math>, the Gini coefficient is:
:<math>G = \frac{1}{2\mu}  \sum\limits_{i=1}^n  \sum\limits_{j=1}^n \, f(y_i) f(y_j)|y_i-y_j|</math>
where
:<math>\mu=\sum\limits_{i=1}^n y_i f(y_i).</math>
If the points with non-zero probabilities are indexed in increasing order <math>(y_i < y_{i+1})</math>, then:
:<math>G = 1 - \frac{\sum_{i=1}^n f(y_i)(S_{i-1}+S_i)}{S_n}</math>
where
:<math>S_i = \sum_{j=1}^i f(y_j)\,y_j\,</math> and <math>S_0 = 0.</math> These formulas are also applicable in the limit, as <math>n\rightarrow\infty.</math>