The fracture strength of hard bricks satisfies an Erlang distribution of order $2$. There are $n \in \mathbb{N}$ fracture strength tests that are carried out.
We consider the statistical product model $(X ,(P_{\theta})_{\theta \in \Theta})$ with $X = (0,\infty)^n$, $\Theta = (0,\infty)$ and densities $f_{\theta}(x_i) = \theta^2x_ie^{-\theta x_i}$ for all $x_i \in (0,\infty)$, $\theta \in \Theta$.
I want to calculate the Likelihood-function.
I have done the following :
\begin{equation*}L_x(\theta)=\prod_{i=1}^nf_{\theta}(x_i)=\prod_{i=1}^n \theta^2x_ie^{-\theta x_i}=\theta^{2n}e^{-\theta \sum_{i=1}^n x_i} \prod_{i=1}^nx_i\end{equation*} Is that correct so far? How could we continue?