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What is the relationship between covariance and independence of random variables?
‘ We’ve said that if random variables are independent, then they have a Covariance of 0; however, the reverse is not necessarily true. That is, if two random variables have a Covariance of 0, that does not necessarily imply that they are independent.
Does covariance measure independence?
Zero covariance – if the two random variables are independent, the covariance will be zero. However, a covariance of zero does not necessarily mean that the variables are independent. A nonlinear relationship can exist that still would result in a covariance value of zero….Covariance.
| x1, | y1 |
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| xn, | yn |
How do you find covariance from independence?
If X and Y are independent variables, then their covariance is 0: Cov(X, Y ) = E(XY ) − µXµY = E(X)E(Y ) − µXµY = 0 The converse, however, is not always true. Cov(X, Y ) can be 0 for variables that are not inde- pendent.
What is positive covariance?
Covariance measures the directional relationship between the returns on two assets. A positive covariance means that asset returns move together while a negative covariance means they move inversely.
Is the covariance matrix the same as the variance matrix?
The matrix is also often called the variance-covariance matrix, since the diagonal terms are in fact variances. By comparison, the notation for the cross-covariance matrix between two vectors is
Is the covariance of an independent variable 0?
But if they are independent, their covariance must be 0. I could not think of any proper example yet; could someone provide one? Easy example: Let X be a random variable that is − 1 or + 1 with probability 0.5. Then let Y be a random variable such that Y = 0 if X = − 1, and Y is randomly − 1 or + 1 with probability 0.5 if X = 1.
Which is the best example of covariance and independence?
Some other examples, consider datapoints that form a circle or ellipse, the covariance is 0, but knowing x you narrow y to 2 values. Or data in a square or rectangle.
How is a pseudo-covariance matrix defined for complex random vectors?
For complex random vectors, another kind of second central moment, the pseudo-covariance matrix (also called relation matrix) is defined as follows. In contrast to the covariance matrix defined above Hermitian transposition gets replaced by transposition in the definition.