In probability theory and statistics, the **covariance** between two real-valued random variables *X* and *Y*, with expected values *E*(*X*) = μ and *E*(*Y*) = ν is defined as:

*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

*X*and

*Y*are independent, then their covariance is zero. This follows because under independence, E(X·Y) = E(X)·E(Y). The converse, however, is not true: it is possible that

*X*and

*Y*are not independent, yet their covariance is zero.

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:

*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.