Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume that $X$ and $Y$ are independent real-valued random variables on $(\Omega,\mathscr{F},P)$ such that $P_X,P_Y=\dfrac{1}{2}(\delta_1+\delta_{-1})$. Show that $X,Y$ and $Z=X\cdot Y$ are pairwise independent but not independent.
Well, I'm already struggling to show their pairwise independence. I'm not quite sure how I can get $P_{XY}$ from $P_X,P_Y$ to start with.