Computing covariance for independently distributed uniform variables $X$, $Y$, and $Z$

Question

Let $X$, $Y$, and $Z$ ~ Uniform(0,1).

How does one set up the computation for $\textrm{Cov}(X^2Y,Y^2Z)$?

Answer 1

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.