Covariance

From Academic Kids

This article is not about the physics topic, covariant transformation, nor about the mathematics example for groupoids, covariance in special relativity, nor about parameter covariance in object-oriented programming.

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

<math>\operatorname{cov}(X, Y) = \operatorname{E}((X - \mu) (Y - \nu)), \,<math>

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

<math>\operatorname{cov}(X, Y) = \operatorname{E}(X Y) - \mu \nu. \,<math>

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

<math>E(X \cdot Y)=E(X) \cdot E(Y)=\mu\nu<math>.

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:

<math>\operatorname{cov}(X, X) = \operatorname{var}(X)\,<math>
<math>\operatorname{cov}(X, Y) = \operatorname{cov}(Y, X)\,<math>
<math>\operatorname{cov}(cX, Y) = c\, \operatorname{cov}(X, Y)\,<math>
<math>\operatorname{cov}\left(\sum_i{X_i}, \sum_j{Y_j}\right) = \sum_i{\sum_j{\operatorname{cov}\left(X_i, Y_j\right)}}\,<math>

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

<math>\operatorname{cov}(X, Y) = \operatorname{E}((X-\mu)(Y-\nu)^\top).\,<math>

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:Covariāncia su:Kovarian

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