The meaning of covariance

The meaning of covariance

Reproduced in: http://bbs.mathchina.com/cgi-bin/topic.cgi?forum=5&topic=14444 (thanks to the original author)

In probability theory, two random variables X and Y are related to each other in roughly the following 3 cases:

When the joint distribution of x, Y is like that, we can see that there are roughly: the larger the X, the greater the X, the smaller the Y, the less the case we call " positive correlation ".

When the joint distribution of x, Y is like that, we can see that there are generally: the larger the x y instead the smaller, the smaller the x, the larger the case, and we call it " negative correlation ".

When the joint distribution of x, Y is like that, we can see: Neither x is larger, y is larger, nor is x larger y but smaller, so we call it " not relevant”。

How can these 3 related situations be expressed in a simple number?

In the plot area (1), there is X>ex, y-ey>0, SO (X-ex) (Y-ey) >0;

In the plot area (2), there is X<ex, y-ey>0, so (X-ex) (Y-ey) <0;

In the plot area (3), there is X<ex, y-ey<0, so (X-ex) (Y-ey) >0;

In the area (4) of the figure, there is X>ex, y-ey<0, so (X-ex) (Y-ey) <0.

when X with theYPositive Correlation, most of their distributions are in the region (1) and (3), a small portion of the region (2) and (4 ), so on average, there is E(x-ex) (Y-ey) >0 .

when  x &NBSP; Y 2 4 1 3 (X-ex) (Y-ey) <0 &NBSP; .

when X is not related to Y, they are distributed in regions (1) and (3), with the region ( 2) and (4) in the distribution almost as much, so on average, there (X-ex) (Y-ey) =0 .

So, we can define a digital feature that represents X, Y correlations, which is covarianceCoV (X, Y) = E (X-ex) (Y-ey).

when CoV (x, y) >0 , it indicates that x and y Positive Correlation ;

when   CoV (X, Y) <0 X and Y ;

when CoV (X, Y) =0 when it indicates X with the Y not relevant .

This is the meaning of covariance.

The meaning of covariance