Suppose that $X_1,\dots,X_n$ is an iid random sample from a Poisson distribution with mean $\theta$. I would like to prove that there exists no unbiased estimator of $\frac{1}{\theta}$.
To do so, I will let $\bar\theta(X)$ be an estimator of $\frac{1}{\theta}$. Then, I'd like to equate the expectation of $\bar\theta(X)$ and $\frac{1}{\theta}$ $$E\left[\bar\theta(X)\right] = \sum_{x=0}^{\infty}\bar\theta(x)P(X=x)$$
Now, my problem is that I don't know what to do next. I can write the probability $P(X=x)$ out but how do I continue from there? I have no information of $\bar\theta$ which is inside the sum. Any ideas?