# Estimation Theory - Basic Question

I'm just starting estimation theory for my DSP course and I'm a bit confused about the classic example given everywhere which aims to show that two unbiased estimators can have different variances.

I'm just wondering how they did that step that I have a question mark beside. I understand intuiviely that the average of x[n] will be A but I can't mathematically figure that out with the E[] and also what happens to 1/N. If someone could just explain that step in detail.

Another Question: What's the difference between A(hat) and E[A(hat)], I mean shouldn't A(hat) give you A anyways? Then what's the point of the E[A(hat)]?

The expectation of $$x[n]$$ is
$$E(x[n])=E(A+w[n])=E(A)+E(w[n])=A+0=A\tag{1}$$
because $$A$$ is deterministic and the noise $$w[n]$$ is assumed to have a mean of zero. So in that formula you sum up $$N$$ terms that are equal to $$A$$, and then you divide that sum by $$N$$, leaving you with the value $$A$$.
As for the expectation of the estimated value $$\hat{A}$$, note that $$\hat{A}$$ is a random variable because it necessarily depends on the data $$x[n]$$, which are random, so computing its expected value (and its variance) makes sense.
• @AlfroJang80: You have $E(w[n])=0$ and that's true for any $n$, also for $n=0$. Note that you consider the zeroth element of all possible realizations of the noise process. – Matt L. Nov 12 '18 at 8:15