a)derive the likelihood estimator of the geometric function $\displaystyle f(x;\theta)=\theta(1-\theta)^{x-1}, x=1,2,3$

let $\displaystyle x_{1},,,,x_{n}$ represent number of attacks,,

b)determine the cramer-rao lower bound for an unbiased estimator of $\displaystyle \Theta$

I have some answers for part a) which gave the MLE as $\displaystyle \frac{n}{\sum x_i}=\frac{1}{\overline{x}}=\frac{1}{E(x)}$ which would work for part b) as it is unbiased...for it to be unbiased the proof is that either MSE(estimate)=var(estimate) or BIAS(estimate)=0 implies E(estimate)=0. hence $\displaystyle E(\frac{1}{E(x)}-\Theta)

=\frac{1}{\frac{1}{\theta}}-\Theta)=0$

however I cannot see how this estimator $\displaystyle \frac{1}{E(x)}$ was obtained.

My working is as follows andI obtained something completely different while the official working gave the anser above: any offerings as to how they did it?

$\displaystyle l(\Theta)=\log\Theta^n(1-\Theta)^{\sum x_{i}-n}$

$\displaystyle =(\sum x_{i}-n)log(1-\Theta)+n\log(\Theta)$

$\displaystyle l'(\Theta)=\frac{\sum x_{i}-n}{1-\Theta}+\frac{n}{\Theta}=0$

$\displaystyle =\Theta(\sum x_{i}-n)+n(1-\Theta)=0$

$\displaystyle =\Theta(\sum x_{i}-n)+n-n\Theta=0$

$\displaystyle =\Theta(\sum x_{i}-2n)=-n$

$\displaystyle \Theta=\frac{=n}{\sum x_{i}-2n}$

$\displaystyle =\frac{n}{2n-\sum x_{i}}$

which is clearly biased