Conditional expectation of an unbiased and sufficent statistic

Question

I'm having some trouble understanding the following equality from my course book.

Some background:

Let $X_1,...,X_n$ be n independent Bernoulli r.v's with unknow parameter $\theta$. $X_1$ is an unibased estimator for $\theta$ and $T(X) = \sum_{i=1}^n X_i$ is a sufficent statistic.

Now to derive an estimator with a smaller variance one can use Rao-Blackwell's theorem and calculate the conditional expectation of the unbiased estimator and the sufficent statistic.

The following is written in my book which I do not understand.

$E_\theta (X_1|\sum_{i=1}^n X_i =t) = P_\theta (X_1 =1|\sum_{i=1}^n X_i = t)$

I tried to compute using some conditional expectation properties but I feel like the course book is skipping a lot of steps or that I might be missing something. Also, why is $X_1 = 1$ in the probability function?

I would appreciate if anyone could explain to me what is going on here.

Answer 1

Since $X_1$ bernoulli, it can only take on values 0,1. Explicitely write out the expectation. For computing the probability, you have to use a combinatorics argument. It's something like this: given that out of n coins, t are heads, there is a total of nCt ways this could happen. Now, say the first coin was a head. Then there are (n-1)C(t-1) ways to have t heads out of n coins with the first being head. Take ratio and you should get $\frac{t}{n}$.

This is reasonable because given that you've seen t heads out of n, your natural guess at the true probability of heads is $\frac{t}{n}$.

Answer 2

Note that $$\mathbb E_{\theta}\left(X_1\mid \sum_{i=1}^n X_i=t\right)=1\cdot P_{\theta}\left(X_1=1\mid \sum_{i=1}^n X_i=t\right)+0\cdot P_{\theta}\left(X_1=0\mid \sum_{i=1}^n X_i=t\right)=$$$$P_{\theta}\left(X_1=1\mid \sum_{i=1}^n X_i=t\right)$$

Observe that under condition $\sum_{i=1}^n X_i=t$ exactly $t$ of the $X_i$ give value $1$ and that all $X_i$ have equal probability to do that.

So the probability that $X_1$ is one of them is $\frac{t}n$.

This tells us that: $$P_{\theta}\left(X_i=1\mid \sum_{i=1}^n X_i=t\right)=\frac{t}n$$

Actually this together justifies the conclusion that: $$\mathbb E_{\theta}(X_1\mid \sum_{i=1}^n X_i)=\frac1n \sum_{i=1}^n X_i=\frac1nT(X)$$(so does not depend on $\theta$).

Does this help?