Compute $Cov(X,Y)$

Question

Let $X: U(-1,1)$ and $Y=X^2$. Compute $\operatorname{Cov}(X,Y)$.

$\operatorname{Cov}(X,Y)=E[XY]-E[X]E[Y]$.

$E[X]= \frac{1}{2}(-1+1)=0$ so $\operatorname{Cov}(X,Y)=E[XY]$, but how can I calculate $E[XY]$?

Answer 1

By definition of $Y$, $$ \mathbb{E}[XY] = \mathbb{E}[X^3]\,. $$ Now, you could compute this explicitly: $$ \mathbb{E}[X^3] = \int_{-1}^1 x^3 f_X(x) dx = \frac{1}{2}\int_{-1}^1 x^3dx \,. $$ However, it suffices to note that $X$ is a symmetric r.v. to see that its odd moments are zero, and thus $\mathbb{E}[XY] =0$ (i.e., $X$ and $Y$, while clearly not independent, are uncorrelated).

Answer 2

\begin{align} E[XY]=E[X^3] = \frac12 \int_{-1}^1x^3 \, dx = 0 \end{align}