Editing Covariance (section)

===Discrete random variables===
If the (real) random variable pair <math>(X,Y)</math> can take on the values <math>(x_i,y_i)</math> for <math>i = 1,\ldots,n</math>, with equal probabilities <math>p_i=1/n</math>, then the covariance can be equivalently written in terms of the means <math>\operatorname{E}[X]</math> and <math>\operatorname{E}[Y]</math> as
<math display="block">\operatorname{cov} (X,Y) = \frac{1}{n}\sum_{i=1}^n (x_i-E(X)) (y_i-E(Y)).</math>

It can also be equivalently expressed, without directly referring to the means, as<ref>{{cite conference |author=Yuli Zhang |author2=Huaiyu Wu |author3=Lei Cheng |title=Some new deformation formulas about variance and covariance|book-title=Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012)|date=June 2012 |pages=987–992}}</ref>
<math display="block"> \operatorname{cov}(X,Y) = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \frac{1}{2}(x_i - x_j)(y_i - y_j) = \frac{1}{n^2} \sum_i \sum_{j>i} (x_i-x_j)(y_i - y_j). </math>

More generally, if there are <math>n</math> possible realizations of <math>(X,Y)</math>, namely <math>(x_i,y_i)</math> but with possibly unequal probabilities <math>p_i </math> for <math>i = 1,\ldots,n</math>, then the covariance is
<math display="block">\operatorname{cov} (X,Y) = \sum_{i=1}^n p_i (x_i-E(X)) (y_i-E(Y)).</math>

In the case where two discrete random variables <math>X</math> and <math>Y</math> have a joint probability distribution, represented by elements <math>p_{i,j}</math> corresponding to the joint probabilities of <math>P( X = x_i, Y = y_j )</math>, the covariance is calculated using a double summation over the indices of the matrix:

<math display="block">\operatorname{cov} (X, Y) = \sum_{i=1}^{n}\sum_{j=1}^{n} p_{i,j} (x_i - E[X])(y_j - E[Y]).</math>