Consider the random variables $Y,\epsilon_1, \epsilon_2$. $\epsilon\equiv (\epsilon_1, \epsilon_2)$ has cumulative distribution function (cdf) $F$.
Let $u_1, u_2$ be some known real-valued parameters.
Suppose $Y$ can take values $\{0,1,2\}$ with the following probabilities $$ (\diamond) \text{ } \begin{cases} \mathbb{P}(Y=1)=\mathbb{P}(\epsilon_1\geq -u_1, \epsilon_1-\epsilon_2\geq u_2-u_1)\\ \mathbb{P}(Y=2)=\mathbb{P}(\epsilon_2\geq -u_2, \epsilon_1-\epsilon_2\leq u_2-u_1)\\ \mathbb{P}(Y=0)=\mathbb{P}( \epsilon_1\leq -u_1, \epsilon_2\leq-u_2)\\ \end{cases} $$ (I haven't done a precise distinction between weak and strict inequalities as I assume that $F$ is continuous)
To understand better I have pictured an example
with $u_1\equiv 3$ and $u_2\equiv -1$. $\mathbb{P}(Y=1)$ is the probability of the pink area, $\mathbb{P}(Y=2)$ is the probability of the green area, $\mathbb{P}(Y=0)$ is the probability of the blue area.
Question: I want to re-write these probabilities using one joint cdf only (not necessarily $F$). Is there a way to do that?
My thoughts:
The tricky part is that
these probabilities depend on different random variables: $\mathbb{P}(Y=1)$ on $\epsilon_1, \epsilon_1-\epsilon_2$; $\mathbb{P}(Y=2)$ on $\epsilon_2, \epsilon_1-\epsilon_2$; $\mathbb{P}(Y=0)$ on $\epsilon_1, \epsilon_2$
the cdf's of $\begin{pmatrix} \epsilon_1\\ \epsilon_1-\epsilon_2 \end{pmatrix}$, $\begin{pmatrix} \epsilon_2\\ \epsilon_1-\epsilon_2 \end{pmatrix}$, $\begin{pmatrix} \epsilon_1\\ \epsilon_2 \end{pmatrix}$ do not seem to me expressible as marginals of a common joint cdf
My idea is to define $$V\equiv \begin{pmatrix} \epsilon_1\\ \epsilon_2\\ \epsilon_1-\epsilon_2 \end{pmatrix} $$ with cdf $G$ and use $G$ as common cdf to re-write those probabilities. With the help of the picture above, $$ \mathbb{P}(Y=1)= $$ $$\mathbb{P}( \epsilon_1\geq -u_1, \epsilon_2\leq -u_2)+ \mathbb{P}(\epsilon_1\geq -u_1, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\geq u_2-u_1) $$ $$ = \mathbb{P}( \epsilon_1\geq -u_1, \epsilon_2\leq -u_2,\epsilon_1-\epsilon_2\geq u_2-u_1)+ \mathbb{P}(\epsilon_1\geq -u_1, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\geq u_2-u_1) $$ where the second equality comes from observing that $$\epsilon_1\geq -u_1, \epsilon_2\leq -u_2 \rightarrow \epsilon_1-\epsilon_2\geq u_2-u_1$$
Similarly, $$ \mathbb{P}(Y=2)= $$ $$\mathbb{P}( \epsilon_1\leq -u_1, \epsilon_2\geq -u_2)+ \mathbb{P}(\epsilon_1\geq -u_1, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\leq u_2-u_1) $$ $$ = \mathbb{P}( \epsilon_1\leq -u_1, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\leq u_2-u_1)+ \mathbb{P}(\epsilon_1\geq -u_1, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\leq u_2-u_1) $$
where the second equality comes from observing that $$\epsilon_1\leq -u_1, \epsilon_2\geq -u_2 \rightarrow \epsilon_1-\epsilon_2\leq u_2-u_1$$
Lastly, $$ \mathbb{P}(Y=0)= $$ $$\mathbb{P}( \epsilon_1\leq -u_1, \epsilon_2\leq -u_2, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\leq u_2-u_1)+ \mathbb{P}( \epsilon_1\leq -u_1, \epsilon_2\leq -u_2, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\geq u_2-u_1) $$ $$ = \mathbb{P}( \epsilon_1\leq -u_1, \epsilon_2\leq -u_2, \epsilon_2\geq -u_2,\epsilon_1-\epsilon_2\leq \infty) $$ All these probabilities can be re-written using $G$. Is this correct?