# Math Help - The Covariance of Two Random Variables

1. ## The Covariance of Two Random Variables

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.

2. Originally Posted by atalay
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...

$X$ takes values $\{-1,0,1\}$ with probabilities $\{\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\}.$

Let $Y = X^2.$

$Cov(X,Y) = EX^3 - EXEX^2 = 0$

Since, $EX = EX^3 = 0.$

But clearly $X$ and $Y$ are dependent.