Suppose that $X_1, \ldots ,X_n$ form a random sample from a geometric distribution with parameter $0<\theta<1$ whose PMF is $p(x) = θ(1−θ)^x$,for $x \in \{0,1,2,\ldots\}$. It is known that this distribution has mean $E[X_i]=\frac{1−\theta}{\theta}$ and variance Var$(X_i) = \frac{1−\theta}{\theta^2}$, for $i \in \{0,1,2,\ldots\}$.
I found that $Y:=\sum x$ is the complete and sufficient statistic for $\theta$ and that the MLE for $\theta=\frac{n}{Y+n}$.
I need to find the unique MVUE for Var$(x)$. so far I have that unbiased estimator for $\frac{1}{\theta}=\frac{Y+n}{n}$.