Mean of the sampling distribution of the OLS estimator Suppose b represents the OLS estimator, and B the true coefficient in the regression model without intercept
y = Bx + u. 

Under certain assumptions b is unbiased so that 
E[b | X] = B. 

Suppose that I want to have an estimator of  
E[b | X].

The estimator of B is b. Hence it should be the estimator of this conditional mean. Then 
Est. E[b | X] = b

where Est. stands for estimator. What does this tell me? The OLS estimate I obtain with the sample data at hand gives me the estimate of the mean of the sampling distribution of b. So we assume that the sample data at hand is so typical that it produces a b that is an estimate of the mean of the sampling distribution of the estimator. Is this correct? 
 A: Your surprise might stem from a reasoning in a different direction, why is $\hat \beta$ not only estimate for $\beta$ but also for the mean of it's own sampling distribution $\mu_{\hat\beta}=E(\hat \beta)$.
But you always get (by default) for any random variable $X$ with finite mean, that a particular value $x$ sampled from the distribution of $X$ is an unbiased estimate for $E(X)$.


*

*a single $b$, drawn from the distribution for $B$ is an unbiased estimator for $E(B)$ 

*if $E(B)=\beta$, then $b$ is (also) an unbiased estimate for $\beta$
The property 1 is: not some additional rule to 2, not a consequence of 2, and neither a consequence of OLS estimation. 
You always have 1, and sometimes 2. 
Your line of thought goes like we have 2, but also 1, why is that? The answer is a bit trivial: since you always have 1.

Where:


*

*$B$ is a variable and more specifically refers to the population (that is all instances of $B$)

*$b$ is a specific draw from $B$, it  refers to a sample (in contrast to the population)

*$\beta$ is a parameter. It is not directly measured. For instance, the slope in a particular relationship/function/model.

*$\hat \beta$ is an estimate for $\beta$ based on a sample.

A: Following your notation and your particular example ... "given certain assumption we have that" 
E[b | X] = B

for the law of iterated exeptaction it follows that:
E[b] = B

$b$ is your ESTIMATOR for $B$ and it is a random variable because its value depends on the sample you draw. It's centered around the $B$ that is its expected value (the mean). Every time you draw a sample the value o $b$ is different but the mean of $b$, $E(b)$  is equal to $B$ .
The mean of the sampling distribution of $b$ is just $B$ as you can read in the equality above.
