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? 
 A: 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.
For example, let $X$ be some real-valued random variable. Then $X$ and $X^2$ are uncorrelated, but clearly not generally independent.
A: If X and Y are Jointly Gaussian and X and Y are uncorrelated then X and Y are independent
A: 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.
A: 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).
A clear description is given on this Wikipedia page:

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.

