Conditional entropy for product of two random variables

Question

$X, Y, Z$ are discrete random variables, and $X$ and $Z$ are independent. We define:

$Y=XZ$.

what can be said about the following quantities:

$H(Y)$ and also $H(Y|X)$?

Since $X$ and $Z$ are independent, can we argue that $H(Y|X) = H(Z|X) = H(Z)$?

Answer 1

Indeed we can. Given $X$, $X$ is a constant, so $H(Y\mid X)=H(XZ\mid X)$ is the entropy of a constant times $Z$, which is just the entropy of $Z$. Also $H(Z\mid X)=H(Z)$ since $Z$ and $X$ are independent.