Does Covariance Remain Unchanged Under Variable Transformations?

[PAHV]

Let X1 and Y1 be two random variables. We have Cov(X1,Y1) = 0. Does this extend to any transformation X2 = g(X1) and Y2 = g(Y1), such that Cov(X2,Y2)? Here, g is a continuous function. For example, if we set X2 = X1^2 and Y2 = Y1^2. Do we then from Cov(X1,Y1) = 0 that Cov(X1^2,Y1^2) = 0?

 

[D H]

No. For example, if X1 and Y1 are uncorrelated but not independent, then your X2 and Y2 may have a non-zero correlation.

 

[Stephen Tashi]

For example, try random variables X and Y with the joint distribution given by:

P(X=-2,Y=3) = 1/4
P(X=-1,Y=2) = 1/4
P(X=1,Y=2) = 1/4
P(X=2,Y=3) = 1/4

and transform by g(w) = w^2