Question

If the covariance of two variables is 0, are they necessarily independent?

Answer 1

No, if X and Y have pairs of values say :
(- 3, 9),(- 2 ,4), (- 1, 1) , ( 0, 0), (1, 1), (2, 4),(3,9) their covariance is zero. But they are not independent, but related; as Y = `X^2`

Explanation:

Cov (X, Y) = (1/n)`sum` xy - `barx` `bary` .
=(1/7) `xx` 0 - 0 `xx` 4 = 0 as
`sum` xy = -3`xx` 9 + (-2)`xx` 4 +( - 1) `xx` 1 + 0`xx` 0 + 1`xx` 1 +2`xx` 4
+3`xx` 9 = - 27 - 8 - 1 + 0 + 1 + 8 + 27 = 0.
Similarly `sum` x = 0 `=>` `barx` = 0