Exponential Distribution

Continuous Distribution.
X~Exp ($\lambda$) : the time between events of a Poisson Process with rate $\lambda$. 
 

Example) about MLE, Pivot and N-P Lemma (Reference: University of Toronto STA355, Final 2013 Q1)
Suppose that $X_{1}, X_{2},...X_{n}$ are independent exponential random variables with density $f(x;\lambda)=\lambda\cdot\exp(-\lambda\cdot x)$ for $x \geq 0$, $\lambda > 0$ 
a) Find the MLE of $\lambda$, and find the limiting distribution of $\sqrt{n}(\hat{\lambda_{n}}-\lambda)$?
b) A pivot for $\lambda$ is $2\lambda \sum_{i=1}^{n}X_{i}\sim \chi^2_{2n}$. Show how you can use this pivot to construct a CI for $\lambda$. 
c) $H_{0}: \lambda =1$ vs. $H_{1}: \lambda > 1$ Suing test statistic $T=2\sum_{i=1}^{n}X_{i}$.
  For an alpha level test, for what values of T would reject $H_{0}$?
Solution??!!