Solution
= Solution
For the overlapping products, factor using the independence of $T,U,V$:
$$
\operatorname{Cov}(TU,UV)=\mathbb E[TU^2V]-\mathbb E[TU]\mathbb E[UV]
=1\cdot2\cdot1-1\cdot1=1.
$$
This is the <covariance of overlapping independent products>: the shared $U$ contributes its second moment. In contrast, $TU$ and $VW$ use disjoint independent factors and are themselves independent, giving
$$
\boxed{\operatorname{Cov}(TU,UV)=1,\qquad\operatorname{Cov}(TU,VW)=0.}
$$