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Let $X_1, X_2..... X_n$ be a random sample from a Bernoulli population with parameter $p$.

A sufficient statistic is $\sum_{i=1}^{n}X_i$. If we define $$ U(X_1,X_2,\ldots,X_n)= \begin{cases}1/2n &\text{if }X_1+X_2=1 \cr 0 &\text{otherwise}\cr \end{cases} $$ is an unbiased estimator of $p(1-p)/n$

Using Rao Backwell theorum,

$\delta(t)=$P(X1+X2=1/ $\sum_{i=1}^{n}$ Xi)

E($\delta(t)$)=p(1-p)/n

Then, $\delta(t)$ is UMVUE of p(1-p)/n.

Is this correct?

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    $\begingroup$ For a self-study question, you need to pursue the resolution of the question further than simply dropping a thought. For instance applying Rao-Blackwell should be within your reach since you have an unbiased estimator AND a sufficient statistic. $\endgroup$
    – Xi'an
    Jun 29 '20 at 11:54
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    $\begingroup$ What I mean is that without further input from you it is not regular on this forum to provide full answers to self-study questions. I thus repeat my request that you produce more details on your attempt within the text of the question and not as additional comments. $\endgroup$
    – Xi'an
    Jun 29 '20 at 13:30

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