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?
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?
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.