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In probability theory and statistics, covariance is a measure of the joint variability of two random variables. [ 1 ] The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables.
In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. [ 1 ] [ 2 ] Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways.
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z ( x ) on a domain D , a covariance function C ( x , y ) gives the covariance of the values of the random field at the two ...
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector.
In probability theory, the law of total covariance, [1] covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
The probability density function of a complex random variable is defined as () = ... The covariance between two complex random variables , is defined as [3] ...
For infinite sequences of exchangeable random variables, the covariance between the random variables is equal to the variance of the mean of the underlying distribution function. [10] For finite exchangeable sequences the covariance is also a fixed value which does not depend on the particular random variables in the sequence.