Q$1$: Assuming that $x_1,x_2,$dots,$x_n$ be i$.$i$.$d$.$ from Exponential$()$ with pdf :

$f(x$;$)=\{\begin{array}{cc}\frac{1}{}{e}^{-\frac{x}{}},& x>0\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

a$)$ Find the most powerful test for testing: $H_0$:$=7$ Vs$.H_1$:$=3$

b$)$ At significance level $=0\mathrm{.}05.$ Find a EXACT likelihood ratio test $($LRT$)$ for testing: $H_0$:$=3$ Vs$.H_1$:$3$

c$)$ At significance level $=0\mathrm{.}05.$

Find an approximation for likelihood ratio test $($LRT$)$ for testing: $H_0$:$=3$ Vs$.H_1$:$3$