Rao Blackwell theorem on Bernoulli distribution

Question

I am currently doing statistics homework and we just covered Rao blackwell theorem. The homework has 3 parts $X_1, \dots , X_n \sim \operatorname{Bernoulli}(p)$ :

1. Give an unbiased estimator
2. Give a sufficient statistics
3. Give a more sufficient estimator than in 1 with Rao Blackwell theorem.

I found :

1. Sample mean
2. Sum of $X_i$ 's from $1$ to $n$

And I need help with the 3rd one. As I understood I need to take the conditional probability of 1 and 2. But I am not sure about how to find it.

Answer 1

1. $\hat{p}=x_1$, hence $\mathbb{E}x_1 = p $.
2. $T(X) = \sum_{i=1}^n x_i$.
3. $$\tilde{p} = \mathbb{E}[x_1|T(X)=y] = \frac{\mathbb{P}( x_1 = 1, \sum_{i=2}^n=y-1) }{\mathbb{P}(\sum_{i=2}^nx_i = y)} = \frac{p\binom{n-1}{y-1}p^{y-1}q^{n-y}}{\binom{n}{y}p^{y}q^{n-y}} = \frac{y}{n}$$ i.e., $$ \tilde{p}^{RB} = \frac{1}{n}\sum_{i=1}^nX_i $$