# If two random variables have a correlation of $0$, do they have to be independent?

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?

## 1 Answer

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.