[Math] Distribution of $X_{1:n} – \eta$ (Two parameter exponential distribution)

Question

$\text{Consider a random sample of size n},\, X_i\sim EXP(1, \eta)$

Note that $X\sim EXP(\theta,\eta)$ is the two parameter exponential distribution with pdf $f(x)=\frac{1}{\theta}e^{\left(\frac{-(x – \eta)}{\theta}\right)}$

Find the distribution of $Q = X_{1:n} – \eta$

I got only as far as that $X_{1:n}$ is the maximum likelihood estimator for $\eta$

Answer 1

$$ P\{X_{1:n}-\eta\le x\}=1-P\{X_{1:n}>x+\eta\}=1-[P\{X_1> x+\eta\}]^n=1-e^{-nx} $$