# UMVUE of a Bernoulli distribution

Suppose that $$(X_1,X_2, \ldots, X_n)$$ be a random sample from Bernoulli (p). Find the UMVUE of

$$\tau(p)= (1-p)+ e^{2}p$$

My approach:

I know $$T(X) = \sum_{i=1}^n X_i$$ is a complete sufficient statistic for the Bernoulli distribution. And by Lehmann_Scheffe theorem,

$$\phi(T)= \mathbb{E}(W(X)\mid T(X))$$

where W(X) is an unbiased estimator. Can one define $$W(X)= (1-X_1)+ e^{2}X_1$$?

How do I continue from here?

• What is your $W$? You have stated the theory without showing any work. – StubbornAtom Apr 7 at 6:37
• I have re-edited the post to include the definition of W. I have no idea how to go about it. That's why I am asking for help. – Lady Apr 7 at 14:17
• If $e^2$ is just a constant, $E\left[W\mid T\right]=E\left[(e^2-1)X_1+1\mid T\right]=(e^2-1)E\left[X_1\mid T\right]+1$. Now you have to deduce the distribution of $X_1\mid T$. – StubbornAtom Apr 7 at 15:08