Say we have two random variables, $X$ and $Z$ that are independent. Then let $W=a+bXZ$ be the random variable that is a function of both $X$ and $Z$, $a$ and $b$ are just scalar constants. Then are $X$ and $W$ independent? How about $Z$ and $W$?
This is just something I thought of, not sure how I would go about proving (or disproving) it.