$(20$ points$)$ Suppose that we have two independent random samples: $x_1,$dots,$x_n$ are

exponential $(),$ and $Y_1,$dots,$Y_n$ are exponential $().$

$($a$)$ Find the LRT of $H_0$:$=$ versus $H_0$:$.$

$($b$)$ Show that the test in part $($a$)$ can be based on the statistic $T=\frac{{\displaystyle \underset{?}{\overset{?}{}}}{x}_i}{{\displaystyle \underset{?}{\overset{?}{}}}{x}_i+{\displaystyle \underset{?}{\overset{?}{}}}{Y}_i}.$

$($c$)$ Find the distribution of $T$ when $H_0$ is true.