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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 expectation operator.
This is equivalent to the following formula which is commonly used in actual 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.
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