Let $X_1, \ldots, X_n$ be i.i.d r.v. with the p.d.f.
$$ f(x; \theta_1, \theta_2) = \begin{cases} \frac{1}{\theta_2} \exp\left(-\frac{x - \theta_1}{\theta_2}\right), & \text{if } x > \theta_1, \\ 0, & \text{otherwise}, \end{cases} $$
where $\theta_1 \in \mathbb{R}$, $\theta_2 > 0$ are unknown parameters.
Using that $T := (\bar{X},X_{(1)})$ (where $X_{(1)} = \min_{1 \leq i \leq n} X_i$) is a complete and sufficient statistic for this model, check whether each of the statistics $$Y_1 := \frac{X_1 - X_2}{\max_{1 \leq i \leq n} X_i},\quad Y_2 := \frac{X_2 - X_3}{S}$$ is independent from $T$.
I have already proven $Y_2$ is independent using the Basu Theorem, the p.d.f. is in the location-scale family, and $Y_2$ is location-scale invariant. I am not sure how to approach $Y_1$ though. Can I please have some help with this question?