Consider a random sample of size n from a distribution with pdf

$\displaystyle f(x;q)=(Ln(q))^x/qx!$ if $\displaystyle x=0,1,...$ and $\displaystyle q>1$and $\displaystyle 0$ otherwise.

a)Find a complete sufficient statistic for$\displaystyle q$

b)Find the maximum likelihood estimator of $\displaystyle q$

c)Find the Cramer-Rao Lower Bound for $\displaystyle q$

d)Find the Uniformly Minimum Variance Unbiased Estimator for $\displaystyle Ln(q)$

e)Find the Uniformly Minimum Variance Unbiased Estimator for $\displaystyle (Ln(q))^2$

(f) Find the Cramer-Rao Lower Bound for $\displaystyle (Ln(q))^2$

for part a) can be writen as$\displaystyle 1/qx!e^{xLn(Ln(q))}$which shows that this is regular exponent. and by Theorem, we know $\displaystyle S=\Sigma X$ is complete suff stat. But how do we find the MLE of q?
for the part b) Find the maximum likelihood estimator of . i found that MLE of$\displaystyle q=e^{\overline{X}}$ where$\displaystyle \overline{X}=\Sigma X/n$. and now i am trying to do part c)Find the Cramer-Rao Lower Bound for . A theorem tells me that (i hope this is correct) to find CRLB of $\displaystyle q$, find $\displaystyle Var(e^{\overline{X}})$.