Computing covariance for independently distributed uniform variables $X$, $Y$, and $Z$ Let $X$, $Y$, and $Z$ ~ Uniform(0,1).
How does one set up the computation for $\textrm{Cov}(X^2Y,Y^2Z)$?

Using the definition of the covariance and the independence,
\begin{align*}
\operatorname{Cov}[X^2Y,Y^2Z]
&=\operatorname E[X^2Y^3Z]-\operatorname E[X^2Y]\operatorname E[Y^2Z]\\
&=\operatorname EX^2\operatorname EY^3\operatorname EZ-\operatorname EX^2\operatorname EY\operatorname EY^2\operatorname EZ.
\end{align*}
Hence, we need to calculate the first three moments of the uniform distribution.
