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Unbiased expression?

  1. Jul 26, 2007 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    [tex]
    \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}
    [/tex]

    3. 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)
     
  2. jcsd
  3. Jul 26, 2007 #2
    A statistic [itex]\tau(x_1,x_2,...,x_n)[/itex] is said to be unbiased for a parameter [itex]\theta[/itex] if [itex]E[\tau(x_1,x_2,...,x_n)]=\theta[/itex].

    It is just a definition.

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

    The expectation of [itex]\hat{p}[/itex] is precisely p. If it so happened that [itex]E[\hat{p}]=p-2[/itex] then [itex]\hat{p}[/itex] would not be an unbiased estimator for p, it would be an unbiased estimator for p-2.
     
    Last edited: Jul 26, 2007
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