Let be distributed i$.$i$.$d$.$ with probability density function

$($a$)$

$3$ points possible $($graded$)$

Let denote the log likelihood. Find the critical point of $.($The critical point is unique because KL divergence is definite.$)$

$($If applicable, enter barX$\_$n for and bar$($X$\_$n$^2)$ for $.)$

Critical point of is at

unanswered

Find the second derivative of $.$ Your answer should be a function of and the data $.$

$($Do not evaluate at the critical point at this stage.$)$

$($If applicable, enter Sigma$\_$i$($X$\_$i$)$ for and and Sigma$\_$i$($X$\_$i$^2)$ for $.)$

unanswered

Using the second derivative test, is the critcal point you obtain above a global maximum, a global minimum, or neither of in the domain $?$

global maximum

global minimum

neither

unanswered

What can you conclude about the maximum likelihood estimator for $?$

$($There is no answer box for this question.$)$