Let $X_1,\dots ,X_n$ be i.i.d CRVs from an exponential distribution with PDF $$f(x\mid \mu)=\frac{1}{\sigma_0}\exp\left(-\frac{x-\mu}{\sigma_0}\right)~1_{x\geq \mu}~~\text{where }\mu,\sigma_0\in\mathbb{R}~\text{are known}$$
(a) Show that $Y_1=\frac{2}{\sigma_0}(X_1-\mu)$ follows a $\chi_2^2$ distribution.
(b) Find $\mathsf E\left(X_{(1)}\sum_{i=1}^n(X_i-X_{(1)})\right)$ (hint: Basu's theorem)
(c) Show that $\frac{2}{\sigma_0}\sum_{i=1}^n\left(X_i-X_{(1)}\right)$ follows a $\chi^2_{2n-2}$ distribution.
While solving part (c) of the above problem, I was trying to equate the moment generating functions of distributions of LHS and RHS. MGF of chi square is known. However, I am facing problem in calculating the MGF of 1st order statistic. Lower limit in the integral is $\mu$, but I cannot decide the upper limit; since it must be smaller than the 2nd order statistic .So, a dependency occurs.
Is there any other way to prove part (c) ? If not, how to calculate the MGF of X(1) ?
