Consider a random variable $T$ with probability density

$f(t)=\{\begin{array}{ccc}\frac{1}{2\sqrt[\phantom{2}]{t}}\text{e}\text{x}\text{p}& (-\frac{\sqrt[\phantom{2}]{t}}{})\text{f}\text{o}\text{r}& t>0\\ 0& \text{f}\text{o}\text{r}& t0\end{array}$

where $$ is an unknown model parameter. Moreover, we have the following i$.$i$.$d$.$ samples:

$\frac{t_1}{11300},\frac{t_2}{5000},\frac{t_3}{4300},\frac{t_4}{8500},\frac{t_5}{7900}$

Derive the log likelihood function and the maximum likelihood estimate $($in general and for the

specific sample$).$