Mathematical covariance formula of senior one.
The variance between two different parameters is covariance. If two random variables X and Y are independent of each other, then E[(X-E(X))(Y-E(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 [(x-e (x)) (y-e (y)) is called covariance of random variables x and y, and it is denoted as COV(X, y), that is, COV(X, y) = E[(X-E(X))(Y-E(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)
Covariance has the following relationship with expected value:
COV(X，Y)=E(XY)-E(X)E(Y).
Properties of covariance:
( 1)COV(X，Y)=COV(Y，X)；
(2)COV(aX, bY)=abCOV(X, y), (a, b are constants);
(3)COV(X 1+X2，Y)=COV(X 1，Y)+COV(X2，Y).
As can be seen from the definition of covariance, COV(X, X)=D(X), COV(Y, Y)=D(Y).
Covariance, as a quantity describing the degree of correlation between X and Y, plays a certain role in the same physical dimension, but the same two quantities adopt different dimensions, which makes their covariances show great differences in value. To this end, the following concepts are introduced:
definition
? XY=COV(X，Y)/？ D(X)？ D(Y) is called the correlation coefficient of random variables X and Y. ..
definition
What if? XY=0, then x and y are said to be uncorrelated.
Namely. The necessary and sufficient condition for XY=0 is that COV(X, Y)=0, that is, uncorrelated and covariance is zero equivalent.
theorem
Settings? XY is the correlation coefficient of random variables x and y, then there are
( 1)∣? XY∣？ 1;
(2)∣? The necessary and sufficient condition for XY∣= 1 is that P{Y=aX+b}= 1, (a, b are constants, a? 0)
definition
Let x and y be random variables. If e (x k), k= 1, 2, ... exists, it is called the K-order origin moment of X, or K-order moment for short.
If e {[x-e (x)] k}, k= 1, 2, ... exists, it is called the k-order central moment of X.
If e (x ky l), k, l= 1, 2, ... exists, it is called the k+l 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 mixed central moment of x and y.
Obviously, the mathematical expectation of X is that E(X) is the first-order origin moment of X, the variance D(X) is the second-order central moment of X, and the covariance COV(X, Y) is the second-order mixed central moment of X and Y. ..