simple covariance homework question

Question

any help is greatly appreciated. I am insecure about whether and how to use the covariance formula for this basic question.

Suppose X is a random variable with E[X]=E[X^3]=0. Suppose that Y=X^2 is another random variable.

a) What is cov[X,Y]? b) Is X independent of Y?

Answer 1

$cov [X,Y]=EXY-EX EY=EX^{3}-EXEX^{2}=0-0=0$.

$X$ and $Y$ are independent iff $X=\pm c$ with probaility $1/2$ each for some $c$. This is because independence of $X$ and $Y$ implies that $Y$ is independent of itself so it is a constant.

Answer 2

Hint for the first part \begin{eqnarray*} Cov(X,Y) =E[XY]-E[X]E[Y]. \end{eqnarray*} Hint for the second part: What does $Cov[X,Y]=0$ tell us about how $X$ & $Y$ are correlated?