Covariance is a term usually associated with random variables. Covariance is a measure of how the change in value of one random variable changes the value of the other random variable. If the increase in value of one random variable increases the value of the other or the decrease in value of one random variable decreases the value of the other, then the covariance is positive. ON the other hand, if the increase in value of one random variable decreases the other, or if the decrease in value of one random variable increases the other, then the covariance is negative .It gives the linear relationship between the two random variables.
An assignment another important factor apart from the sign of the covariance is the magnitude of the value, be it high positive low positive or high negative low negative. The magnitude often tells the strength of the relationship between the two random variables. Calculation of covariance is not a difficult calculation as it involves some expectation terms. Suppose X and Y are to random variables. Then covariance between X and Y is given by: COV(X, Y) =E ((X-E(X)) (Y-E(Y))
The above equation can be modified :COV(X,Y)=E(XY-XE(Y)-YE(X)+E(X)E(Y))=E(XY)-E(X)E(Y).This formula is used very often for calculating covariance .When the random variables X and Y are independent , then E(XY)=E(X) * E(Y).In such cases, the covariance between X and Y is 0.Covariance is also used for calculating the correlation coefficient. The correlation coefficient is a dimensionless quantity. Its value lies between -1 and 1. The correlation coefficient is the ratio of covariance and square root of the product of the variance of X and Variance of Y. Thus the Covariance is less than or equal to square root of the product of the variance of X and Variance of Y because the correlation coefficient is within -1 to 1.
There are several properties of covariance
· COV(X,X)=E(X^2)-E(X)^2
=V(X)
Thus Covariance of random variable with itself is just the variance of the random variable.
· COV(X, a) =0, where a is a constant. Thus the covariance of a random variable with a constant is always 0.
· COY(X, Y) =COV(Y, X) which means the order of the random variable doesn’t matter.
· COV (aX, bY) =abCOV(X, Y). The proof the property is given below:
COV (aX, bY) =E (aXbY)-E (aX) E (bY)
=ab(E(XY)-E(X)*E(Y))
=ab COV(X,Y)
· The above property described the shift of scale property. We now discuss the shift of origin property: COV(X+a,Y+b)=COV(X,Y)
COV(X+a,Y+b)=E((X+a)(Y+b))-E(X+a)E(Y+b)
=E(X *Y+Xb+Ya+ab)-E(X)* E(Y)-E(X)* b-E(Y) *a-a *b
=E(XY) - E(X) *E(Y)
= COV(X,Y)
· Var(X+Y)=V(X)+V(Y)+2cov (X,Y)
When X and Y are independent, then covariance between X and Y IS 0. In such cases, V (X+Y)= V (X) + V (Y)
We have stated that two independent variables have zero covariance but is the converse true? Does zero covariance imply that the random variables are independent? Well infect it does not. If the covariance is zero it just means that there is no linear relationship between the two random variables but they can be dependent on each other in some other way.
So a wise argument is that there might be a dependency of X on Y but they can’t be regarded as independent.
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