## Interview Question: Lack of Correlation

Sample Question #107 (probability theory)

For two random variables X and Y, when can zero correlation between the two imply independence?

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### 7 Responses to Interview Question: Lack of Correlation

1. Brett says:

As you probably know, in general zero correlation does not imply independence, as independence between two random variables is a stronger condition than non-correlation.

However, exceptions do occur.  The best known exception is when both X and Y are normally distributed. In that case, if corr(X, Y) = 0, then X and Y must also be independent.

2. quant says:

sorry, you are wrong. You can have X, Y normally distributed, uncorrelated, but dependent! The best known example is for X, Y to by jointly normal, which is a difference!!!

3. Wu Chao says:

The best known example is for X, Y to by jointly normal, which is a difference!!!what’s this?

4. quant says:

for example, independent normal random variables are jointly normal. For them it holds that they are uncorrelated and independent. More generally, jointly normal random variables will be correlated if and only if they are dependent.As I already said, you can set up two normal random variables that will be uncorrelated, but will not be independent!!! (they will necessarily not be jointly normally distributed ) Go check out your favorite probability book …

5. Brett says:

anonymous, here’s what you said "jointly normal random variables will be correlated if and only if they are dependent." But that contradicts your first post!

The normal distribution is special in many ways, one of which is its equivalency between noncorrelation and independence.

6. quant says:

what did I say in the first post? The best known example that has the properties you mention, ie. correlation only if dependence, is for X, Y to by jointly normal.You write:"The normal distribution is special in many ways, one of which is its equivalency between noncorrelation and independence. "- which is WRONG!!!

7. Zhe says:

The only thing u need know: correlation contain only linear information, not whole.