Exercise :
Let $X_1, \dots, X_n$ be a random sample from the Exponential Distribution with unknown parameter $\theta$.
(i) Find a sufficient and complete statistics function $T$, for $\theta$.
(ii) Using without proof known formulas, find the distribution of $T$.
Attempt :
(i) The p.d.f. for the sample is given as :
$$f(x;\theta) = \begin{cases} \theta e^{-\theta x}, & x \geq 0 \\ 0, & x<0 \end{cases}$$
Thus $f(x;\theta) = \theta e^{-\theta x}\mathbb{I}_{[0,+\infty]}(x) $ which belongs to the Exponential Family of Distributions, thus the function:
$$ T = \sum_{i=1}^nx_i$$
is a sufficient and complete statistics function for $\theta$.
(ii) Question : How would one proceed with finding the distribution of $T$ now ?
