Here is the problem I've been working on for a while now:
Let $X_1,..., X_n$ be a random sample of size n from a binomial distribution with pdf $f(x|p)=C^m_xp^x(1-p)^{m-x}$, where $m \in Z^+$ and $p\in(0,1)$. Find: a. The UMVUE for p. b. Show that the estimator in (a) converges in probability to p as n → ∞. (c) Establish both the asymptotic and the exact distribution of the estimator in (a).
I found that the UMVUE for p is $Y=\frac{1}{m}\frac{1}{n}\sum_iX_i$. Then I said that by the WLLN we know that $\bar{X}$ converges in probability to $mp$ as $n\rightarrow \infty$. Further, we know that Y is a function of $\bar{X}$, thus, Y converges in probability to p as $n\rightarrow \infty$.
For part (c) this is what I have so far: By CLT we know that $\frac{\bar{X}-mp}{\frac{\sqrt{mp(1-p)}}{\sqrt{n}}} \rightarrow N(0,1)$ in distribution. Rearranging the expression we get: $\frac{Y-p}{\frac{m\sqrt{mp(1-p)}}{\sqrt{n}}}\rightarrow N(0,1)$, so $Y\rightarrow N(p,\frac{m^3p(1-p)}{n})$ in distribution. I am not sure if this is correct since I don't have a good understanding of asymptotic distribution... nor how to find exact distribution of an estimator. If somebody could give me some detailed help with it I would really appreciate it.