Consider a random sample $x_1,x_2,$dots,$x_n$ from a distribution

with probability function:

$f(x,)=(1-{)}^x{I}_{\{0,1,2,\text{d}\text{o}\text{t}\text{s}\}}(x)$

a$.$ Find the uniformly most powerful $($UMP$)$ test for testing the

hypothesis $H_o$:${}_0$ against $H_1$:$>{}_0.$

b$.$ Find the likelihood ratio $($LR$)$ test for testing the hypothesis

$H_o$:$={}_o$ vs $H_1$:${}_o$ and compare $($to see whether they

are equivalent or not$)$ with the union intersection test $($UIT$)$

constructed from the two one$-$sided UMP tests.

c$.$ By inverting the UIT or the LR test, find a $95\%$ exact CI for

$$ and $\frac{1}{}.$

d$.$ Find an exact and asymptotic $($large sample$)95\%$ CIs for $\frac{1-}{}.$