Covariance & Correlation

Covariance has a upper and lower limit.

-SD(X) SD(Y) <=  Cov(X,Y) <= SD(X) SD(Y)

If a normal random variable is scaled, covariance gets affected. It counters the intuition that dependent relationships are unaltered due to scaling of normal random variables.

Therefore, covariance, as a dependence measure fails in such cases. Thus we need a dependence measure which is unaffected by the scaling (is dimensionless) and that is why correlation exists.