The likelihood function for a single parameter $\backslash$theta $,$ based on n iid observations,

X $=($X$1,...,$ Xn$),$ satisfies the following equation

$$ log L$($X; $\backslash$theta $)$

$=$ a$(\backslash$theta $)($Y $\backslash$theta $)$

$\backslash$theta

where a$(\backslash$theta $)>0$ is some differentiable function of $\backslash$theta only and Y is a statistic depending

on data X$.$ Assume the usual regularity conditions $($e$.$g$.,$ intergral and derivative can

be interchanged$).$

$($a$)$ Find a sufficient statistic $($other than the original X$)$ for $\backslash$theta $.$ Justify your

answer.

$($b$)$Find the Maximum Likelihood Estimate $($MLE$)$ of $\backslash$theta $.$ Is the MLE unbiased

for $\backslash$theta $?$ Justify your answer.

$($c$)$ Find the variance of the MLE $($you may assume the order of integration

and differentiation can be interchanged in your derivation$)$