Conditional expectation of binomial distribution

Question

Let $X_1$, $X_2$,..., $X_n$, $n>2$, be a random sample from the binomial distribution $b(1,\theta)$ and let $Y_2=(X_1+X_2)/2$.

I know how to show that $Y_1$=$X_1+X_2$ +$\dots$+ $X_n$ is a complete sufficient statistic for $\theta$ and I can find out $Y_1/n$ is the MVUE of $\theta$.

My question is how can I determine $\mathbb{E} \{ Y_2 |Y_1=y_1 \}$. I have trouble dealing this type of conditional probability problem.

Thanks.

Answer 1

Let $S_n=X_1+\dots +X_n$. Then

$$\mathbb{E}[X_1|S_n]=\mathbb{E}[X_2|S_n]=...=\mathbb{E}[X_n|S_n]$$

Then $\mathbb{E}[X_1|S_n]+\cdots+\mathbb{E}[X_n|S_n]=\mathbb{E}[S_n|S_n]=S_n$ so that $$\mathbb{E}[X_i|S_n]=\frac{S_n}{n} \text{, }i=1,...,n$$

and $$\mathbb{E}[Y_2|S_n]=\frac{1}{2}\{\mathbb{E}[X_1|S_n]+\mathbb{E}[X_2|S_n]\}=\frac{S_n}{n}$$