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  1. If we have two random variables $X$ and $Y$ that have non-zero standard deviations and their correlation is $0$, are $X$ and $Y$ independent?

  2. Say we have two random variables $X$ and $Y$ with a correlation of $0$, and $X$ and $Y$ have non-zero standard deviations. Is $\operatorname{var}(X + Y) = \operatorname{var}(X) + \operatorname{var}(Y)$?

For 1), I feel it's possible that $X$ and $Y$ do not necessarily have to be independent but still have a correlation of $0$. I'm thinking that if they could have positive and negative correlations for certain parts of the distribution but the total correlation equates to $0$. However, I'm having trouble translating my thoughts into concrete equations.

For 2), I know that through simplification:

$$\operatorname{var}(X + Y) = \operatorname{var}(X) + \operatorname{var}(Y) + 2(E[XY] − E[X]E[Y]).$$

Since $X$ and $Y$ are not independent, we can't cancel out the last term automatically. However, we know the correlation is zero and standard deviations are non-zero, does this mean that the covariance is zero, meaning we can cancel out the last term in the above expression? Is my logic valid for this problem?

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You're logic is correct in the second part. For the first part there's a whole thread of counterexamples here: https://stats.stackexchange.com/questions/85363/simple-examples-of-uncorrelated-but-not-independent-x-and-y.

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