# UMVUE of Bernoulli random variables

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?

• 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. Jun 29 '20 at 11:54
• 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. Jun 29 '20 at 13:30