Assume $X$ to be standard normal random variable, and define $Y$ as$$Y=\begin{cases}X,&\text{if }⌊X⌋\text{ is even}\\-X,&\text{if }⌊X⌋\text{ is odd}\end{cases}.$$
I am trying to show that $X$ and $Y$ are mutually completely dependent. For this I want to find the support of the copula of $(X,Y)$: $$ C(u,v)=\mathbb{P}( X \leq F^{-1}(u), Y \leq G^{-1}(v)),$$ and conclude by finding the probability mass concentrated on some locus. Here $F(x) = \Phi(x)$ is the standard normal distribution of $X$, and $G(y)$ is the distribution of $Y$.
However, I am not sure how to technically tackle the $Y$, that is, how to properly split it between cases odd/even integer values.
My approach:
I am adding an excerpt from another question of mine (which is linked) with my approach. If my work is correct, I worked out that $$G(y)=\frac{1}{2}[1 + F(y) - F(-y)].$$
Further,\begin{align*}
C(u,v) &= \mathbb{P}(X \leq F^{-1}(u), Y \leq G^{-1}(v))\\
&= \frac{1}{2}\left[\mathbb{P}(X \leq F^{-1}(u), X > -G^{-1}(v)) + \mathbb{P}(X \leq F^{-1}(u), X \leq G^{-1}(v)) \right]\\
&=\frac{1}{2}\left[\mathbb{P}(X \leq \min\{F^{-1}(u), G^{-1}(v)\}) + \mathbb{P}(X \in [-G^{-1}(u), F^{-1}(v)]) \right]\\
&= \frac{1}{2}\left[ v - F(G^{-1}(v)) + F(\min\{F^{-1}(u), G^{-1}(v)\}) \right]. \tag{1}
\end{align*}
It seems I might be able to untangle this as a function of $(u,v)$ if I find $F(G^{-1}(v))$. However, here I'm not too sure that the usual approach of finding the inverse works.
$$ y = G(G^{-1}(y)) =\frac{1}{2}[1 + F(G^{-1}(y)) - F(-G^{-1}(y))] $$ $$ 2y -1 = F(G^{-1}(y)) - F(-G^{-1}(y)) $$ $$ F^{-1}(2y -1) = F^{-1}\left( F(G^{-1}(y)) -F(-G^{-1}(y)) \right) \tag{2}$$
And it seems I am stuck here. I am thinking that it is enough to simplify $(1)$ to some convenient form, here my idea is that finding the expression for $G^{-1}(y)$ would help, but I got stuck at $(2)$.
Maybe another approach is better?
Would it be easier to show that $\mathbb{P}(X = g(Y))= 1$, where $g$ is some function of $Y$? But I still need to find the support of $C$ for following exercises.
Would appreciate any hints or suggestions on how to proceed!
