$\newcommand{\Cov}{\operatorname{Cov}}$Given :

  • $X$, $Y$, $S$ are independent random variables.
  • $X \sim U(-1,1)$
  • $Y \sim \exp(2)$
  • $S \sim N(4,3^2)$

Find: $$ \Cov((X^2-1)Y + X^3S, X) $$

I got to:

\begin{align} & \Cov((X^2-1)Y + X^3S, X) \\[10pt] = {} & \Cov(X^2Y,X) - \Cov(Y,X) + \Cov(X^3S,X) \\[10pt] = {} & \Cov(X^2Y,X) + \Cov(X^3S,X) \end{align}

Now I'm having trouble calculating the two covariences.



For $\text{Cov}(X^3S,X)$, we need $E(X^4S)-E(X^3S)E(X)$. It is easy to verify that $E(X)=0$.

So we want $E(X^4S)$, which by independence is $E(X^4)E(S)$. Finally, $E(X^4)$ is easily calculated, it is $\frac{1}{10}$, and $E(S)$ is known.

The calculation of $\text{Cov}(X^2Y,X)$ is similar but a little simpler.

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