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 ?

  • $\begingroup$ Relevant thread: math.stackexchange.com/q/655302/321264. $\endgroup$ – StubbornAtom Jun 6 '18 at 10:40
  • 1
    $\begingroup$ Use MGFs to show that $T$ has a gamma distribution with shape parameter $n.$ Wikipedia on 'gamma distribution' might be helpful. $\endgroup$ – BruceET Jun 7 '18 at 4:33

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