I am confused on the implications of imposing strictly positive density for continuous random variables.
Take the continuous real-valued random variables $X,Y,Z,W$ mutually independent.
Suppose that each random variable has strictly positive density on $\mathbb{R}$ (in my mind I picture as an example normally distributed random variables). Does this imply that
(a) any linear function of $X,Y,Z,W$ (e.g. $X+Y-W$) has strictly positive density on $\mathbb{R}$?
(b) the function $t(X,Y)+Z-W$ has strictly positive density on $\mathbb{R}$ for any function $t:\mathbb{R}^2\rightarrow \mathbb{R}$?
Any example violating (a),(b) would help me to clarify the issue.
I am tempted to say yes to (a), (b) but I am not sure.