"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

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

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.