Hi

@kevolution Per your formula, σ(12) = σ(1)* σ(2)*ρ(1,) which is classically important, zero correlation implies zero covariance, and indeed just as

*logical converse* is true: zero covariance implies zero correlation. The key semantic distinction (at least for our purposes) is

**dependence versus independence**. Independence is a key statistical assumption, proven when joint probability(X,Y) = P(X)*P(Y), but it's commonly violated in practice. If variables are not independent, they are (to some degree) dependent. But dependence encompasses an entire set of different relations, only one of which is linear dependence (i.e., correlation or covariance). My diagram below is maybe not the best Venn diagram ever

but something like this:

So my Venn tries to convey:

- Non-zero correlation or covariance =
*linear *dependence
- If independence --> correlation and covariance are zero (outside the circle is zero correlation including all of the yellow square)
- If zero correlation, then we can't infer dependence or independence (could be either). I hope that's helpful!

... so independent variables do imply zero correlation (and zero covariance), but it is not true that zero correlation (or zero covariance) implies independence because there can be

**non-linear dependence**. I hope that helps!

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