Covariance
In probability theory as well as statistics, covariance is the measure of the joint variability of two random variables. whether the greater values of one variable mainly correspond with the greater values of the other variable, as living as the same holds for the lesser values that is, the variables tend to show similar behavior, the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, that is, the variables tend to show opposite behavior, the covariance is negative. Theof the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not easy to interpret because it is not normalized in addition to hence depends on the magnitudes of the variables. The normalized explanation of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation.
A distinction must be presents between 1 the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and 2 the sample covariance, which in addition to serving as a descriptor of the sample, also serves as an estimated usefulness of the population parameter.
Properties
The variance is a special issue of the covariance in which the two variables are identical that is, in which one variable always takes the same value as the other:: p. 121
If 
, 
, 
, and 
are real-valued random variables and 
are real-valued constants, then the following facts are a consequence of the definition of covariance:
For a sequence 
of random variables in real-valued, and constants 
, we have
A useful identity to compute the covariance between two random variables 
is the Hoeffding's covariance identity:
where 
is the joint cumulative distribution function of the random vector 
and 
are the marginals.
Random variables whose covariance is zero are called covariance matrix is zero in every everyone outside the leading diagonal are also called uncorrelated.
If and are independent random variables, then their covariance is zero.: p. 123 This follows because under independence,
The converse, however, is not loosely true. For example, let be uniformly distributed in ![[-1,1]](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
and permit 
. Clearly, and are non independent, but
In this case, the relationship between and is non-linear, while correlation and covariance are measures of linear dependence between two random variables. This example shows that whether two random variables are uncorrelated, that does not in general imply that they are independent. However, if two variables are jointly ordinarily distributed but not if they are merely individually commonly distributed, uncorrelatedness does imply independence.
Many of the properties of covariance can be extracted elegantly by observing that it satisfies similar properties to those of an inner product:
In fact these properties imply that the covariance defines an inner product over the quotient vector space obtained by taking the subspace of random variables with finitemoment and identifying all two that differ by a constant. This identification turns the positive semi-definiteness above into positive definiteness. That quotient vector space is isomorphic to the subspace of random variables with finite moment moment and intend zero; on that subspace, the covariance is precisely the L2 inner product of real-valued functions on the pattern space.
As a result, for random variables with finite variance, the inequality
holds via the Cauchy–Schwarz inequality.
Proof: If 
, then it holds trivially. Otherwise, let random variable
Then we have