Relation between covariance and uncorrelation/independence

Question

I need to have an explanation... If I have a covariance equal to zero, the random variables are only uncorrelated, or are uncorrelated and independent at the same time?

Thanks

Answer 1

They are uncorrelated but not independent, in general. Conversely, if the variables are Independent they are also uncorrelated.

EDIT 2: If the model is Gaussian, and to be more precise, if $(X,Y)$ are jointly Gaussian, Uncorrelation and independence are the same

Proof:

$$f_{XY}(x,y)=\frac{1}{2\pi\sigma_{X}\sigma_{Y}\sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}[\frac{(x-\mu_{X})^2}{\sigma_{X}^2}-2\rho\frac{(x-\mu_{X})(y-\mu_{Y})}{\sigma_{X}\sigma_{X}}+\frac{(y-\mu_{Y})^2}{\sigma_{Y}^2}]}$$

Letting $\rho=0$ that means incorrelation between $X$ and $Y$ immediately we get

$$f_{XY}(x,y)=f_X(x)f_Y(y)$$

that means also independence.