Hi, Mona,
If two random variables are independent, they have zero covariance; so, the correlation and regression coefficients will be zero too. Independence implies uncorrelatedness, but not the converse. Two variables can have a non-linear or non-monotonic association and be zero correlated. Uncorrelatedness only implies independence if the joint distribution of X and Y is bivariate normal.
I hope this helps,
Braulio.
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