(COMP233) Probability and Statics - Expectation, Variance and Covariance

Covariance

: ​Covariance and Correlation, both describe the relationship between two numerical variables.

 

How to get a covariance value between X and Y?

1) Find the deviation from X to 

x. (Y and 

y)   ------------>  (x-

x)  and (y-

y)

2) multiply (x-

x)  and (y-

y) 

3) Add the values, find the sum 

4) divide by n-1 

         -> WHY did we divide by n-1, not just n?

            : " Degrees of Freedom" - We can't calculate the covariance for only the first value of X and Y. 

                                                              It could become a moot(고려할 가치 X) calculation! 

 

Cov[X, Y ]  =  E[(X − E[X])(Y − E[Y ])]

Using an argument similar to that for variance, we can rewrite this as, 

Cov[X, Y ] = E[(X − E[X])(Y − E[Y ])] 

                 = E[XY − XE[Y ] − Y E[X] + E[X]E[Y ]] 

                 = E[XY ] − E[X]E[Y] 

 

*** If X and Y are independent, then cov(X,Y) = 0 ***

If X and Y are independent, then E[f(X)g(Y )] = E[f(X)]E[g(Y )].

-> It does not mean that cov(X,Y)=0 equals to X and Y are independent !!!

*** If cov(X,Y)=0 , we say that X and Y are uncorrelated ***

 

 

reference : Khan academy, Xiao, Y. comp 233(concordia uni.), Zedstatics(youtube)