Let $X$ and $Y$ be independent random variables, each taking the values $-1$ or $1$ with probability $1/2$, and let $Z = XY$. Show that $X$, $Y$, and $Z$ are pairwise independent. Are they independent?
MY ATTEMPT
To begin with, let us determine the distribution of $Z$: if $X = -1$ and $Y = -1$ or $X = 1$ and $Y = 1$, then $Z = 1$. If $X = -1$ and $Y = 1$ or $X = 1$ and $Y = -1$, then $Z = -1$. Since we considered all possible cases, here is the distribution of $Z$: \begin{align*} p_{Z}(1) = \textbf{P}(Z = 1) = \textbf{P}(X = -1)\textbf{P}(Y = -1) + \textbf{P}(X = 1)\textbf{P}(Y = 1) = \frac{1}{2} \end{align*}
On the other hand, we have \begin{align*} p_{Z}(-1) = \textbf{P}(Z = -1) = \textbf{P}(X = -1)\textbf{P}(Y = 1) + \textbf{P}(X = 1)\textbf{P}(Y = -1) = \frac{1}{2} \end{align*}
Therefore $Z\sim Bernoulli(1/2)$.
As to the second part of the question, we have \begin{align*} p_{X,Z}(1,1) & = \textbf{P}(Z = 1 \mid X = 1)\textbf{P}(X = 1) = \textbf{P}(XY = 1\mid X = 1)\textbf{P}(X = 1)\\ & = \textbf{P}(Y = 1\mid X = 1)\textbf{P}(X = 1) = \textbf{P}(Y = 1)\textbf{P}(X = 1) = \frac{1}{4} = p_{X}(1)p_{Z}(1) \end{align*}
Analogous reasoning applies to the remaining cases. However, the random variables $X$, $Y$ and $Z$ are not independent, because
\begin{align*} p_{X,Y,Z}(1,-1,1) = \textbf{P}(Z = 1\mid\{X = 1\}\cap\{Y = -1\})\textbf{P}(Y = -1\mid X = 1)\textbf{P}(X = 1) = 0 \end{align*}
My question is: are my results correct? Can it be improved? Any contribution is appreciated. Thanks in advance.