Rao-Blackwell and Cramer-Rao LB comparison Let $X_1, X_2, \dots, X_n$ be a random sample following the Geometric distribution. 
$$
\prod\limits_{i=1}^{n} f(x_i|p) = (1-p)^{\sum\limits_{i=1}^n x_i-n}p^n
$$
Since the pmf of the Geometric distribution is exponential family, the factorization theorem yields that the statistic
$$
T = \sum\limits_{i=1}^n x_i
$$
is sufficient and complete. Then, 
$$
E[T] = E\left[ \sum\limits_{i=1}^n x_i \right] = \sum\limits_{i=1}^n E[x_i] = \frac{n}{p}
$$
Therefore, according to Rao-Blackwell, the Minimum Variance Unbiased Estimator of $\frac{1}{p}$ is
$$
W = \frac{1}{n} \sum\limits_{i=1}^n x_i
$$
Now, Cramer-Rao's Lower Bound:
$$
LB = \frac{\left[\left(\frac{1}{p}\right)'\right]^2}{nI(p)} = \frac{1}{n}\frac{1-p}{p} 
$$
Question: Is $V[W] = LB$  in this specific example? If so, is there a reason why $W$ has the lowest possible variance that has something to do with the geometric distribution?
 A: Pmf of the sample $X_1,X_2,\ldots,X_n$ is 
$$f_p(x_1,\ldots,x_n)=p^n (1-p)^{\sum_{i=1}^n x_i-n}\,\mathbf1_{x_1,\ldots,x_n\in\{1,2,\ldots\}}$$
Therefore, we have
\begin{align}
\frac{\partial}{\partial p}\ln f_p(x_1,\ldots,x_n)&=\frac{n}{p}-\frac{1}{1-p}\left(\sum_{i=1}^n x_i-n\right)
\\&=\frac{n}{p(1-p)}-\frac{1}{1-p}\sum_{i=1}^n x_i
\\&=-\frac{n}{1-p}\left(\frac{1}{n}\sum_{i=1}^n x_i-\frac{1}{p}\right)
\end{align}
Thus the score function can be expressed in the form $$\frac{\partial}{\partial p}\ln f_p(x_1,\ldots,x_n)=k(p)\left(T(x_1,\ldots,x_n)-g(p)\right)\tag{*}$$
, for some statistic $T$ and some parametric function $g(p)$. The condition $(*)$ is the equality condition of the Cramer-Rao inequality, which directly shows here that variance of $T=\frac{1}{n}\sum\limits_{i=1}^n X_i$ attains the Cramer-Rao lower bound for $g(p)=1/p$. And since $T$ is unbiased for $1/p$, this also proves that $T$ is the uniformly minimum variance unbiased estimator of $1/p$. 
This fact is more to do with properties of the canonical exponential family than with geometric distribution in particular. Here is another example.
