# probability and statistics: Does having little correlation imply independence?

Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as independent random variables?

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Good question actually; the answer is not intuitive. – Noldorin Aug 17 '10 at 11:05

Independence of random variables implies that they are uncorrelated, but the reverse is not true. Hence no, such an approximation is not valid (given that information alone).

If X and Y are independent, then they are uncorrelated. However, not all uncorrelated variables are independent. For example, if X is a continuous random variable uniformly distributed on [−1, 1] and Y = X², then X and Y are uncorrelated even though X determines Y and a particular value of Y can be produced by only one or two values of X.

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"X2" there should be $X^2$. Can you edit please? – Nate Eldredge Aug 17 '10 at 13:46
@Nate: Certainly. Copy and paste is all to liable to fail! – Noldorin Aug 17 '10 at 13:54
Thanks Noldrin for this valuable information – dikuve Aug 18 '10 at 6:51

If X and Y are Jointly Gaussian and X and Y are uncorrelated then X and Y are independent

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The correlation coefficient only measures linear dependence of two random variables. So if they depend on each other in a non-linear way, the correlation coefficient will not catch it.

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Thank you for this additional information. – dikuve Aug 18 '10 at 8:51

It depends on what else you know about the relationship between the variables.

If the correlation coefficient is the full extent of your information, then the approximation is unsafe, as Noldorin points out.

If, on the other hand, you have good external evidence that the coefficient adequately captures the level of a small linear relationship (eg, a slight dependence on some third quantity that is not germane to your analysis), then it may well be valid to approximate them as independent for some purposes.

RVs about which you know nothing are useful abstractions -- and this is, after all, the maths site -- but real world data often exist in less of vacuum. If you're analysing in the context of a model, that may help you to work out what approximations you can get away with.

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Thanks for your valuable answer. – dikuve Aug 18 '10 at 8:50