Covariance

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For the physics topic, see covariant transformation; about the mathematics example for groupoids, see covariance in special relativity; for the computer science topic see parameter covariance.

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 can also be written:

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

Intuitively, covariance is the measure of how much two variables vary together (as distinct from variance, which measures how much a single variable varies). If two variables tend to vary together (that is, when one of them is above its expected value, then the other variable tends to be above its expected value too), then the covariance between the two variables will be positive.

On the other hand, if when one of them is above its expected value, the other variable tends to be below its expected value, then the covariance between the two variables will be negative.

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

The converse, however, is not true: if X and Y have covariance zero, they need not be independent.

The units of measurement of the covariance cov(X, Y) are those of X times those of Y. By contrast, correlation, which depends on the covariance, is a dimensionless measure of linear dependence.

Random variables whose covariance is zero are called uncorrelated.

[edit] Properties

If X, Y are real-valued random variables and a, b are 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}(aX, bY) = ab\, \operatorname{cov}(X, Y)\,</math>

For sequences X₁, ..., X_n and Y₁, ..., Y_m of random variables, we have

<math>\operatorname{cov}\left(\sum_{i=1}^n {X_i}, \sum_{j=1}^m{Y_j}\right) = \sum_{i=1}^n{\sum_{j=1}^m{\operatorname{cov}\left(X_i, Y_j\right)}}.\,</math>

For a sequence X₁, ..., X_n of random variables, we have

<math>\operatorname{var}\left(\sum_{i=1}^n X_i \right) = \sum_{i=1}^n \operatorname{var}(X_i) + 2\sum_{i,j\,:\,i<j} \operatorname{cov}(X_i,X_j).</math>

[edit] Covariance matrices

For column-vector valued random variables X and Y with respective expected values μ and ν, and m and n scalar components respectively, the covariance is defined to be the m×n 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 related. The correlation is a closely related concept used to measure the degree of linear dependence between two variables.