Let $p \in (0,1)$ and $q=1-p$.
(a) Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,...$ defines a probability distribution for the random variable $X$. (proved)
(b) Given the single observation $X$, the statistic $X$ is sufficient; is $X$ also complete?
Progress: I want to apply the definition of being complete and sufficient statistic, i.e. if $E(X)=0$, then we should have $P(X=0)=1$. Here if $E(X)=0$, then we know the observation is just zero, so the probability is 1.
(c)Determine all unbiased estimators of $p$, given one observation of $X$ from the family above.
Progress: We know $T(X)=\mathbb 1_{\{X=-1\}}$ is an unbiased estimator. I don't know how to decide all the unbiased estimators.
(d) Find the UMVUE of $p$, or prove that it does not exist.