Correlation coefficient
A numerical characteristic of the joint distribution of two random variables, expressing a relationship between them. The correlation coefficient for random variables
and
with mathematical expectations
and
and non-zero variances
and
is defined by
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The correlation coefficient of and
is simply the covariance of the normalized variables
and
. The correlation coefficient is symmetric with respect to
and
and is invariant under change of the origin and scaling. In all cases
. The importance of the correlation coefficient as one of the possible measures of dependence is determined by its following properties: 1) if
and
are independent, then
(the converse is not necessarily true). Random variables for which
are said to be non-correlated. 2)
if and only if the dependence between the random variables is linear:
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The difficulty of interpreting as a measure of dependence is that the equality
may be valid for both independent and dependent random variables; in the general case, a necessary and sufficient condition for independence is that the maximal correlation coefficient equals zero. Thus, the correlation coefficient does not exhaust all types of dependence between random variables and it is a measure of linear dependence only. The degree of this linear dependence is characterized as follows: The random variable
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gives a linear representation of in terms of
which is best in the sense that
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see also Regression. As characteristic correlations between several random variables there are the partial correlation coefficient and the multiple-correlation coefficient. For methods for testing independence hypotheses and using correlation coefficients to study correlation, see Correlation (in statistics).
Correlation coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correlation_coefficient&oldid=46522