Gebruiker:Pjetter/klad: verschil tussen versies

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Pjetter (overleg | bijdragen)
Pjetter (overleg | bijdragen)
Regel 242:
 
:<math>\hat \beta=\frac{\sum_i^{n} x_i y_i}{\sum_i^{n} x_i^2}</math>
 
:<math>B(\widehat{\theta}) = \operatorname{E}(\widehat{\theta}) - \theta</math>
 
:<math>B(\widehat{\theta}) = 0</math>
 
So check if
 
:<math>\operatorname{E}(\widehat{\theta}) = \theta</math>
 
Let us use this on the estimator in a):
 
:<math>\hat \beta=\frac{\sum_i^{n} x_i Y_i}{\sum_i^{n} x_i^2}</math>
 
and take an expectation from both sides:
 
:<math> \operatorname{E}(\widehat{\beta}) = \operatorname{E}(\frac{\sum_i^{n} x_i Y_i}{\sum_i^{n} x_i^2})</math>
 
Now let us fill in the proposed population model into this equation:
 
:<math>\operatorname{Y}_i = \beta x_i + \epsilon_i</math>
 
and you will get:
 
:<math> \operatorname{E}(\widehat{\beta}) = \operatorname{E}(\frac{\sum_i^{n} x_i (\beta x_i + \epsilon_i) }{\sum_i^{n} x_i^2})</math>
 
==Sigma==
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