the following three statements are made in the book that I doubt:1.correlation coefficient is a measure of linear relationship (association) between two random variableswhat is meant by "linear"? are there non-linear assocation?2.if two random variables are independent, the correlation coefficient is zero; but the reverse is falsewhy the reverse is false, are there any implication for a zero correlation coefficient?3.a positive correlation coefficient means the two variables are more 'likely' to go up or down togethera value of 0.5 is more 'likely' than 0.3? how 'likely'? thanks in advance.

1. Suppose that y = x^2. y is perfectly determined by x, but the association is non-linear. If you compute a correlation coefficient from data you can get a negative, zero or positive result depending on what x's you use.2. Zero correlation coefficient means the expected value of x*y equals the expected value of x times the expected value of y. That is often a useful fact to know.3. Your statement is not necessarily correct, although it is usually true. y might usually move in the opposite direction as x, but still have a positive correlation because when both move in the same direction, they do so by a lot. Think instead that if x moves a standard deviations, on average y will move a*correlation standard deviations. If the correlation coefficient between x and y is 0.5, the standard deviation of x is 1 and the standard deviation of y is 3; then if x moves up 2, on average y will move up 3.

>> 2.if two random variables are independent, the correlation coefficient is zero; but the reverse is falsewhy the reverse is false, are there any implication for a zero correlation coefficient? A good example of this is that sin(X) and cos(X) have zero correlation, for a uniform random X. But clearly, sin(X) and cos(X) are not independent. (the implication for zero correlation may have to do with orthogonality in the square sense.)

Hi Dino1019,Hope the following also helps:"2.if two random variables are independent, the correlation coefficient is zero; but the reverse is falsewhy the reverse is false, are there any implication for a zero correlation coefficient?"I guess another example for the case of zero correlation coefficient does not =>independence in random variables X and Y :Let Y = -X Since oppposite signs balance out, covariance is zero which brings zero corralation, but they are dependent processes"3.a positive correlation coefficient means the two variables are more 'likely' to go up or down togethera value of 0.5 is more 'likely' than 0.3? how 'likely'? "I visualize 'likeness' increase in co-movement of random variablesas descreasing of the effect of random disturbances in the relationship structure of the two variables.Please correct me if I am wrong.Regards,asd

Let Y = -X Since oppposite signs balance out, covariance is zero which brings zero corralation, but they are dependent processesThe correlation of X and -X is -1

linear correlation is a basic measure for dependencies. But it cannot tell us everything about risk structures. In addition, correlation is not invariant under transformations of risks!!! It is only defined when variance is finite, which is not always the case due to extreme events. You can also find other dependence measure like rank correlation (invariant under transformations) and copulas.

QuoteOriginally posted by: akimonThe correlation of X and -X is -1 I think the example is supposed to be flip a coin for each X, heads means Y = X, tails means Y = -X. X and Y will have zero correlation, but knowing X can tell you quite a bit about Y.

Building on Aaron's points above, I believe that one way to think about the relationships between uncorelatedness and independence is as follows. Uncorrelatedness obviously requires that the correlation between the levels of X and Y is zero. Independence requires that the correlation between all possible transforms of X and Y are zero (i.e. no matter how you transform x and/or y, the resulting variables still show no correlation).richg.

I think it's even easier to see if you generalize what Aaron said just a little bit. When x is distributed symmetrically then x*abs(x) and -x*abs(x) contribute equally to the expectation you take in calculating correlation (x, abs(x)) so you get correlation(x, abs(x)) =0.