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questions :Covariance theory
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[Member (365WT)]answers [Chinese ]Time :2019-10-11
The variance between two different parameters is the covariance. If two random variables X and Y are independent of each other, then E[(XE(X))(YE(Y))]=0, so if the above mathematical expectation is not zero, Then X and Y must not be independent of each other, that is, there is a certain relationship between them.

definition

E[(XE(X))(YE(Y))] is called the covariance of the random variables X and Y, denoted as COV(X,Y), ie COV(X,Y)=E[(XE(X) )(YE(Y))].

There is the following relationship between covariance and variance:

D(X Y)=D(X) D(Y) 2COV(X,Y)

D(X-Y)=D(X) D(Y)-2COV(X,Y)

The covariance has the following relationship with the expected value:

COV (X, Y) = E (XY) - E (X) E (Y).

The nature of covariance:

(1) COV (X, Y) = COV (Y, X);

(2) COV (aX, bY) = abCOV (X, Y), (a, b is a constant);

(3) COV (X1 X2, Y) = COV (X1, Y) COV (X2, Y).
Defined by covariance, it can be seen that COV(X,X)=D(X), COV(Y,Y)=D(Y).

Covariance, as an amount describing the degree of correlation between X and Y, has a certain effect under the same physical dimension, but the same two quantities use different dimensions so that their covariances show a large difference in value. To introduce the following concepts:

definition

ρXY=COV(X,Y)/√D(X)√D(Y), which is called the correlation coefficient of the random variables X and Y.

definition

If ρXY=0, then X is said to be uncorrelated with Y.

That is, the sufficient and necessary condition for ρXY=0 is COV(X,Y)=0, that is, the irrelevance and the covariance of zero are equivalent.

theorem

Let ρXY be the correlation coefficient between the random variables X and Y, then

(1) ∣ρXY∣≤1;
(2) ∣ρXY∣=1 is a necessary condition for P{Y=aX b}=1, (a, b is a constant, a≠0)

definition

Let X and Y be random variables. If E(X^k), k=1, 2,... exist, it is called k-order origin moment of X, which is called k-order moment.

If E{[X-E(X)]^k}, k=1, 2,... exists, it is called the k-th order central moment of X.

If E(X^kY^l), k, l=1, 2,... exist, it is called the k l-order mixed origin moment of X and Y.

If E{[X-E(X)]^k[Y-E(Y)]^l}, k, l=1, 2,... exists, it is called the k l-order mixed center moment of X and Y.

Obviously, the mathematical expectation of X is E(X) is the first-order origin moment of X, the variance D(X) is the second-order center moment of X, and the covariance COV(X, Y) is the second-order mixed center moment of X and Y.
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