# 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