Q$9.$ Let $x_1,x_2,$cdots,$x_n$ be an independent random sample from a distribution, which has

the density given by

$f(x$;$)=\frac{}{x^{+1}}$ for $x1$

where $>0$ is an unknown parameter.

$($i$)$ Write down the likelihood function of $$ based on $x_1,x_2,$cdots,$x_n.$

$($ii$)$ Find a scalar sufficient statistic for $.$

$($iii$)$ Find the Fisher information about $$ contained in the data $x_1,x_2,$cdots,$x_n,$ i$.$e$.,I_n().$

$($iv$)$ Find the Cramer$-$Rao Lower Bound for estimation of $.$

$($v$)$ What is the MLE of $?$

$($vi$)$ State the asymptotic distribution of the MLE you found in $($v$).$