# Covariance

In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values [itex]E(X)=\mu[itex] and [itex]E(Y)=\nu[itex] is defined as:

[itex]\operatorname{cov}(X, Y) = \operatorname{E}((X - \mu) (Y - \nu)), \,[itex]

where E is the expected value. This is equivalent to the following formula which is commonly used in calculations:

[itex]\operatorname{cov}(X, Y) = \operatorname{E}(X Y) - \mu \nu. \,[itex]

If X and Y are independent, then their covariance is zero. This follows because under independence,

[itex]E(X \cdot Y)=E(X) \cdot E(Y)=\mu\nu[itex].

The converse, however, is not true: it is possible that X and Y are not independent, yet their covariance is zero. Random variables whose covariance is zero are called uncorrelated.

If X and Y are real-valued random variables and c is a constant ("constant", in this context, means non-random), then the following facts are a consequence of the definition of covariance:

[itex]\operatorname{cov}(X, X) = \operatorname{var}(X)\,[itex]
[itex]\operatorname{cov}(X, Y) = \operatorname{cov}(Y, X)\,[itex]
[itex]\operatorname{cov}(cX, Y) = c\, \operatorname{cov}(X, Y)\,[itex]
[itex]\operatorname{cov}\left(\sum_i{X_i}, \sum_j{Y_j}\right) = \sum_i{\sum_j{\operatorname{cov}\left(X_i, Y_j\right)}}\,[itex]

For column-vector valued random variables X and Y with respective expected values μ and ν, and n and m scalar components respectively, the covariance is defined to be the n×m matrix

[itex]\operatorname{cov}(X, Y) = \operatorname{E}((X-\mu)(Y-\nu)^\top).\,[itex]

For vector-valued random variables, cov(X, Y) and cov(Y, X) are each other's transposes.

The covariance is sometimes called a measure of "linear dependence" between the two random variables. That phrase does not mean the same thing that it means in a more formal linear algebraic setting (see linear dependence), although that meaning is not unrelated. The correlation is a closely related concept used to measure the degree of linear dependence between two variables.de:Kovarianz es:Covarianza it:Covarianza no:Kovarians pl:Kowariancja pt:Covariância su:Kovarian

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy