Ex. 10.109- follow Wackerleys solution but expand and explain as much as possible.
Let $X_1, X_2, \dots, X_m$ denote a random sample from the exponential density with mean $\theta_1$ and let $Y_1, Y_2, \dots, Y_n$ denote an independent random sample from an exponential density with mean $\theta_2$.
a. Find the likelihood ratio criterion for testing $H_0: \theta_1 = \theta_2$ versus $H_a: \theta_1 \neq \theta_2$.
b. Show that $\frac{\bar{X}}{\bar{Y}} \sim F(2m, 2n)$. Note the following:
are not so readily available. However, we will show in Exercise 6.46 that if $Y$ has a gamma distribution with $\alpha = n/2$ for some integer $n$, then $2Y/\beta$ has a $\chi^2$ distribution with $n$ degrees of freedom.
