Joint CDF of 2 Unif(0,1) that are based on 3 other Unif(0,1)?

Solution 1:

Note that $X = A - B \pmod 1$ and $Y = C - B \pmod 1$.

Now let $\alpha, \beta \in \mathbb R$ be some fixed numbers. Let $A_{\alpha} = A + \alpha$ and $C_{\beta} = C + \beta$. With these, form $X_{\alpha} = A_{\alpha} - B \pmod 1$ and $Y_{\beta} = C_{\beta} - B \pmod 1$.

Then on the one hand clearly the joint distribution of $(X_{\alpha}, Y_{\beta})$ is equal to that of $(X, Y)$ shifted cyclically $\pmod 1$ over the offset $(\alpha, \beta)$.

On the other hand $A_{\alpha} \pmod 1$, $B$, $C_{\beta} \pmod 1$ are independent and uniform on $[0,1]$. Therefore $(X_{\alpha}, Y_{\beta})$ and $(X, Y)$ have the same distribution. Both observations combined show that this joint distribution is invariant under shifts $\pmod 1$ and is therefore uniform.