Basu's Theorem

Definition

Let $X_1,X_2,\dots ,X_n$ be a Random Sample with PDF $f(x|\theta )$, where $\theta \in \Omega$, $Y$ be a complete sufficient Statistic for $\theta$, and $Z$ be a Ancillary Statistic to $\theta$, then UMP $Y$ and $Z$ are independent.