Invariance for exponential functionals

Let $X_{1},X_{2},…,X_{10}$ be a random sample of size $10$ from an $\operatorname{Exp}(\beta)$ distribution, where $\beta>0$ is an unknown parameter. Let $T=\frac{1}{10}\sum_{i=1}^{10}X_{i}$. Consider the following statements. $P$: $(\log_{e} 2)T$ is a maximum likelihood estimator of median of $X_{(1)}$. $Q$: $e^{-\frac{2}{T}}$ is a maximum likelihood estimator of $P(X_{1}<2)$. Then