Difference between independence and correlation – my understanding

Basically, everyone (or almost everyone) knows that independence and correlation equal to zero are different concepts. More importantly, they don’t have two-way relations, which is if you know two random variables X, Y have zero correlation, you cannot imply that X and Y are independent. However, if X and Y are independent, they will be uncorrelated with zero correlation.

There are lots of ways to illustrate this relation, but I would like to put my understanding into such analysis. 🙂

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