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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?

introvertsdoitbetter
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  • What have you tried so far? Have you written down the definition of $\text{Cov}(X,Y)$ and plugged in $Y=X^2$? Where are you stuck? – angryavian Jan 14 '20 at 00:12
  • I have, I just have trouble reading the the chain of E[X]=E[X^3]=0. It literally means the expected value of X is equal to the expected value of X^3 is equal to zero. So because Y=X^2 and is multiplied by X in the formula you can E[X^3] which is 0, yes? – introvertsdoitbetter Jan 14 '20 at 00:23

2 Answers2

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

$X$ 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.

Kavi Rama Murthy
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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?

Donald Splutterwit
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