"Explaining Unbiased Expression with Probability

[TSN79]

Homework Statement

In an example my book says that the expression bellow is unbiased.
I can't see why this is exactly...

Homework Equations

$$\begin{array}{l} \hat p = \frac{X}{n} \\\\ E(\hat p) = E\left( {\frac{X}{n}} \right) = \frac{1}{n} \cdot E(X) = \frac{1}{n} \cdot (n \cdot p) = p \\ \end{array}$$

The Attempt at a Solution

Could the reason be that the expression comes down to just p, which is simply a probability and we have no better suggestion than to believe that it "hits the target"? (If that didn't make any sense, just ignore it)

 

[ZioX]

A statistic $\tau(x_1,x_2,...,x_n)$ is said to be unbiased for a parameter $\theta$ if $E[\tau(x_1,x_2,...,x_n)]=\theta$.

It is just a definition.

It is important to know that to say that $\hat{p}=\frac{x}{n}$ is unbiased is WRONG. It is unbiased for a particular PARAMETER.

The expectation of $\hat{p}$ is precisely p. If it so happened that $E[\hat{p}]=p-2$ then $\hat{p}$ would not be an unbiased estimator for p, it would be an unbiased estimator for p-2.