Which Fisher information should I use for Cramer-Rao lower bound?

Question

For $X_1,\dots, X_n$ iid sample from $X\sim Bernoulli(p)$. I try to verity that the estimator $\hat{p}=\bar{X}$ (sample mean) is the UMVUE for unknown parameter $p$.

I know that $\hat{p}$ is unbiased. I try to show that $Var[\hat{p}]=CRLB$ (Cramer-Rao lower bound), which is $$ CRLB=\frac{[I'(\theta)]^2}{nI(\theta)}, $$ and $I$ is the Fisher information.

My question is that what is the Fisher information here?

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Note that the score function is $$U(p)=\frac{x-p}{p(1-p)}$$ for one sample.

Also, $$U(p)=\frac{\sum x_i-np}{p(1-p)}$$ for $n$ samples $X_1,\dots, X_n$.

I have two $I(p)$ here. I am not sure if I choose the Fisher information based on the $U(p)$ for one sample or $n$ sample...

$$ I_1=E\left[\frac{(X-p)^2}{(p(1-p))^2}\right]=\frac{1}{p(1-p)} $$ and another one is $$ I_2=E\left[\frac{(\sum X_i-np)^2}{(p(1-p))^2}\right]=\frac{n}{p(1-p)} $$

Answer 1

Cramér-Rao Lower Bound would be of the form $\operatorname{Var}_\theta(T(\mathbf X) ) \geq \mathscr I(\theta)^{-1}. $ For exponential family, $\mathscr I(\theta) =\mathbb E_\theta\left[-\partial^2_\theta \ln f(\mathbf x;\theta)\right],$ which for $X_i\overset{\text{i.i.d.}}{\sim}\mathrm{Ber}(p) $ is

\begin{align}\mathscr I(p) &=\mathbb E_p\left[-\partial^2_p\ln f(\mathbf x;p)\right]\\&=\mathbb E_p\left[-\sum_{i=1}^n\partial^2_p \ln f( x_i;p)\right]\\&=\mathbb E_p\left[-\sum_{i=1}^n\left\{-\frac{x_i}{p^2}+\frac{1-x_i}{(1-p)^2}\right\}\right]\\&=n[p(1-p)]^{-1}.\tag 1\label 1\end{align}

From $\eqref 1,$ it follows the CRLB is $\frac{p(1-p)}n.$