It is known that when random variables X and Y independent, cov(X,Y)=0. However, inverse is not true. Show that, by an example, that we can have cov(X,Y)=0 and X and Y are not independent.
Here is one...
$\displaystyle X $ takes values $\displaystyle \{-1,0,1\}$ with probabilities $\displaystyle \{\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\}.$
Let $\displaystyle Y = X^2.$
$\displaystyle Cov(X,Y) = EX^3 - EXEX^2 = 0$
Since, $\displaystyle EX = EX^3 = 0.$
But clearly $\displaystyle X $ and $\displaystyle Y$ are dependent.