Cov(x, y)=0, then x and y are independent of each other. In fact, the original definition should be E(XY)=E(X)*E(Y). But the conclusion is the same. (For normal distribution only)

A correlation coefficient of 0 does not mean independence, but irrelevance. Irrelevance means that there is no linear relationship, but it does not mean that there is no other relationship. For two-dimensional normal distribution, independence and independence are equivalent. But for other distributions, it is not equivalent.

Y 1 and Y2 are one-dimensional normal distributions, but their joint distribution is two-dimensional normal distribution, so Y 1 and Y2 are independent if they are not related.