Consider a random sample of size $n$ from a two-parameter exponential distribution, $X_i\sim EXP(\theta, \eta)$.
In other words, the density function of $X_i$ is given by:
$$f_X(x;\theta,\eta)=\frac{e^{-(x-\eta)/\theta}}{\theta}~\mathbf 1_{x\gt \eta, \theta>0}$$
Let $\bar X$ be the sample mean and let $X_{(1)}$ be the minimum order statistic.
Show that the conditional pdf of $X_{(1)}$ given $\bar X$ does not depend on $\theta$.
Since I don't know either conditional distribution or the joint distribution, I don't understand how I'm supposed to calculate this PDF. Can someone please help? Thank you.