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

Oct 22, 2015

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 - $\overline{x}$ $\overline{y}$ .
=(1/7) $\times$ 0 - 0 $\times$ 4 = 0 as
$\sum$ xy = -3$\times$ 9 + (-2)$\times$ 4 +( - 1) $\times$ 1 + 0$\times$ 0 + 1$\times$ 1 +2$\times$ 4
+3$\times$ 9 = - 27 - 8 - 1 + 0 + 1 + 8 + 27 = 0.
Similarly $\sum$ x = 0 $\implies$ $\overline{x}$ = 0